Algebraic Algebraic Structures
Compiled from Wikipedia-EJH2013
Contents 1
Abel Ab elia ian n gro group up
1.1 Defini Definitio tionn . . . . . . . . . . . . . . . 1.22 Fac 1. acts ts . . . . . . . . . . . . . . . . . 1.2. 1. 2.11 No Nota tatition on . . . . . . . . . . . 1.2.2 Multi Multipli plicati cation on tabl tablee . . . . . . 1.3 Ex Examp ample less . . . . . . . . . . . . . . . 1.4 Hist Historic orical al rem remarks arks . . . . . . . . . . 1.5 Pr Prope operti rties es . . . . . . . . . . . . . . . 1.6 Fini Finite te abe abelia liann grou groups ps . . . . . . . . . 1.6.1 1.6 .1 Cla Classi ssific ficati ation on . . . . . . . . . 1.6.2 1.6 .2 Au Autom tomorp orphis hisms ms . . . . . . . 1.7 Infin Infinite ite abe abelian lian grou groups ps . . . . . . . . 1.7.1 1.7 .1 To Torsi rsion on gro groups ups . . . . . . . . 1.7.2 Tor Torsio sion-f n-free ree and mix mixed ed grou groups ps 1.7.3 Inv Invaria ariants nts and cla classifi ssificati cation on . . 1.7.4 Ad Additi ditive ve group groupss of rings . . . 1.8 Relatio Relationn to other mathemati mathematical cal topics topics . 1.9 A note on the typog typograph raphyy . . . . . . . 1.10 See also . . . . . . . . . . . . . . . . 1.11 Notes . . . . . . . . . . . . . . . . . 1.12 References . . . . . . . . . . . . . . 1.13 External links . . . . . . . . . . . . . 2
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Categ Cat egor ory y the theor ory y
2.1 Basic Basic con concep cepts ts . . . . . . . . . . . . . . 2.2 App Applic licatio ations ns of Cate Categori gories es . . . . . . . . 2.33 Utitilility 2. ty . . . . . . . . . . . . . . . . . . . 2.3.1 Categorie Categories, s, ob objects, jects, and morphism morphismss 2.3. 2. 3.22 Fu Func ncto tors rs . . . . . . . . . . . . . 2.3.3 Natura Naturall trans transfforma ormation tionss . . . . . 2.4 Categorie Categories, s, ob objects, jects, and morphis morphisms ms . . . . 2.4.1 2.4 .1 Cat Catego egori ries es . . . . . . . . . . . . . 2.4.2 2.4 .2 Mor Morphi phism smss . . . . . . . . . . . .
1 1 1 1 1 2 2 2 2 3 3 4 4 4 4 4 5 5 5 5 6 7
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2.5 Func 2.5 Functo tors rs . . . . . . . . . . . . . . . . . . . . . . . 2.6 Natur Natural al trans transfo formati rmations ons . . . . . . . . . . . . . . . 2.7 Othe Otherr con concep cepts ts . . . . . . . . . . . . . . . . . . . . 2.7.1 Univ Universal ersal construc constructions, tions, limits, and colimits 2.7.2 Equi Equival valent ent cate categori gories es . . . . . . . . . . . . 2.7.3 Furth Further er con concep cepts ts and res results ults . . . . . . . . 2.7.4 Higher-di Higher-dimensi mensional onal categori categories es . . . . . . . 2.8 Hist Historic orical al note notess . . . . . . . . . . . . . . . . . . . 2.99 Se 2. Seee al also so . . . . . . . . . . . . . . . . . . . . . . . 2.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . 2.11 References . References . . . . . . . . . . . . . . . . . . . . . . 2.12 Further reading . . . . . . . . . . . . . . . . . . . 2.13 External links . . . . . . . . . . . . . . . . . . . . 3
CONTENTS
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Fieeld Fi
3.1 Definiti Definition on and ill illustra ustration tion . . . . . . . . . . . . . . . 3.1.1 Firs Firstt ex exampl ample: e: ration rational al num numbers bers . . . . . . . 3.1.2 Second exampl example: e: a field with four eleme elements nts . . 3.1.3 Alte Alternati rnative ve axio axiomati matizati zations ons . . . . . . . . . . . 3.2 Related algebra algebraic ic structures . . . . . . . . . . . . . . 3.2. 3. 2.11 Re Rema mark rkss . . . . . . . . . . . . . . . . . . . 3.33 Hi 3. Hist stor oryy . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Ex Examp ample less . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Rati Rational onalss and alg algebr ebraic aic num numbers bers . . . . . . . 3.4.2 Real Reals, s, comp complex lex num numbers, bers, and p-adic numbers 3.4.3 Cons Construct tructibl iblee numb numbers ers . . . . . . . . . . . . 3.4.4 3.4 .4 Fin Finite ite fie field ldss . . . . . . . . . . . . . . . . . . 3.4.5 Arc Archime himedean dean fie fields lds . . . . . . . . . . . . . . 3.4.6 Fie Field ld of fu functi nctions ons . . . . . . . . . . . . . . . 3.4.7 Local and glo global bal fie fields lds . . . . . . . . . . . . 3.5 Some first theor theorems ems . . . . . . . . . . . . . . . . . . 3.6 Cons Construc tructing ting fie fields lds . . . . . . . . . . . . . . . . . . . . 3.6.1 Clos Closure ure ope operati rations ons . . . . . . . . . . . . . . 3.6.2 Subfie Subfields lds and field extens extensions ions . . . . . . . . . 3.6.3 3.6 .3 Rin Rings gs vs fie field ldss . . . . . . . . . . . . . . . . . 3.6.4 3.6 .4 Ult Ultrap raprod roduc ucts ts . . . . . . . . . . . . . . . . . 3.7 Gal Galoi oiss the theory ory . . . . . . . . . . . . . . . . . . . . . 3.8 Gen General eralizat ization ionss . . . . . . . . . . . . . . . . . . . . 3.8.1 3.8 .1 Exp Expon onent entia iatio tionn . . . . . . . . . . . . . . . . 3.9 App Applic licatio ations ns . . . . . . . . . . . . . . . . . . . . . . 3.10 See also . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . .
9 9 10 10 10 10 10 11 11 11 12 13 13 14
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CONTENTS
3.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.13 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4
Galois group
4.1 4.2 4.3 4.4 4.5 4.6 4.7 5
Definition . . Examples . . Properties . . See also . . . Notes . . . . References . . External links
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Group
5.1 Definition and illustration . . . . . . . . . . . . . . . . 5.1.1 First example: the integers . . . . . . . . . . . 5.1.2 Definition . . . . . . . . . . . . . . . . . . . . 5.1.3 Second example: a symmetry group . . . . . . 5.2 History . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Elementary consequences of the group axioms . . . . . 5.3.1 Uniqueness of identity element and inverses . . 5.3.2 Division . . . . . . . . . . . . . . . . . . . . . 5.4 Basic concepts . . . . . . . . . . . . . . . . . . . . . 5.4.1 Group homomorphisms . . . . . . . . . . . . . 5.4.2 Subgroups . . . . . . . . . . . . . . . . . . . . 5.4.3 Cosets . . . . . . . . . . . . . . . . . . . . . . 5.4.4 Quotient groups . . . . . . . . . . . . . . . . . 5.5 Examples and applications . . . . . . . . . . . . . . . . 5.5.1 Numbers . . . . . . . . . . . . . . . . . . . . 5.5.2 Modular arithmetic . . . . . . . . . . . . . . . 5.5.3 Cyclic groups . . . . . . . . . . . . . . . . . . 5.5.4 Symmetry groups . . . . . . . . . . . . . . . . 5.5.5 General linear group and representation theory . 5.5.6 Galois groups . . . . . . . . . . . . . . . . . . 5.6 Finite groups . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Classification of finite simple groups . . . . . . 5.7 Groups with additional structure . . . . . . . . . . . . . 5.7.1 Topological groups . . . . . . . . . . . . . . . 5.7.2 Lie groups . . . . . . . . . . . . . . . . . . . . 5.8 Generalizations . . . . . . . . . . . . . . . . . . . . . 5.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . .
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5.11 Citations . . . . . . . . . . . 5.12 References . . . . . . . . . . 5.12.1 General references . 5.12.2 Special references . . 5.12.3 Historical references 6
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CONTENTS
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Group theory
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Homomorphism
7.1 Definition and illustration 7.1.1 Definition . . . . 7.1.2 Basic examples . 7.2 Informal discussion . . . 7.3 Types . . . . . . . . . .
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6.1 Main classes of groups . . . . . . . . . . . . . . . 6.1.1 Permutation groups . . . . . . . . . . . . . 6.1.2 Matrix groups . . . . . . . . . . . . . . . . 6.1.3 Transformation groups . . . . . . . . . . . 6.1.4 Abstract groups . . . . . . . . . . . . . . . 6.1.5 Topological and algebraic groups . . . . . . 6.2 Branches of group theory . . . . . . . . . . . . . . 6.2.1 Finite group theory . . . . . . . . . . . . . 6.2.2 Representation of groups . . . . . . . . . . 6.2.3 Lie theory . . . . . . . . . . . . . . . . . . 6.2.4 Combinatorial and geometric group theory . 6.3 Connection of groups and symmetry . . . . . . . . 6.4 Applications of group theory . . . . . . . . . . . . 6.4.1 Galois theory . . . . . . . . . . . . . . . . 6.4.2 Algebraic topology . . . . . . . . . . . . . 6.4.3 Algebraic geometry and cryptography . . . 6.4.4 Algebraic number theory . . . . . . . . . . 6.4.5 Harmonic analysis . . . . . . . . . . . . . . 6.4.6 Combinatorics . . . . . . . . . . . . . . . . 6.4.7 Music . . . . . . . . . . . . . . . . . . . . 6.4.8 Physics . . . . . . . . . . . . . . . . . . . 6.4.9 Chemistry and materials science . . . . . . 6.4.10 Statistical Mechanics . . . . . . . . . . . . 6.5 History . . . . . . . . . . . . . . . . . . . . . . . . 6.6 See also . . . . . . . . . . . . . . . . . . . . . . . 6.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . 6.8 References . . . . . . . . . . . . . . . . . . . . . . 6.9 External links . . . . . . . . . . . . . . . . . . . . 7
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48 48 49 49 50
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CONTENTS
7.4 7.5 7.6 7.7 7.8 7.9 8
7.3.1 Category theory . Kernel . . . . . . . . . . Relational structures . . Formal language theory . See also . . . . . . . . . Notes . . . . . . . . . . References . . . . . . . .
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Ideal
History . . . . . . . . . . . . Definitions . . . . . . . . . . Properties . . . . . . . . . . . Motivation . . . . . . . . . . Examples . . . . . . . . . . . Ideal generated by a set . . . . 8.6.1 Example . . . . . . . 8.7 Types of ideals . . . . . . . . 8.8 Further properties . . . . . . . 8.9 Ideal operations . . . . . . . . 8.10 Ideals and congruence relations 8.11 See also . . . . . . . . . . . . 8.12 References . . . . . . . . . .
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Integral domain
52 52 53 53 53 53 54 54 55 55 55 55 56 57
9.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 9.2 Examples . . . . . . . . . . . . . . . . . . . . . . . 9.3 Non-examples . . . . . . . . . . . . . . . . . . . . . 9.4 Divisibility, prime elements, and irreducible elements 9.5 Properties . . . . . . . . . . . . . . . . . . . . . . . 9.6 Field of fractions . . . . . . . . . . . . . . . . . . . 9.7 Algebraic geometry . . . . . . . . . . . . . . . . . . 9.8 Characteristic and homomorphisms . . . . . . . . . . 9.9 See also . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 References . . . . . . . . . . . . . . . . . . . . . .
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10 Isometry
10.1 Introduction . . . . 10.2 Formal definitions . 10.3 Examples . . . . . 10.4 Linear isometry . . 10.5 Generalizations . .
50 50 51 51 51 51 51 52
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57 57 58 58 58 59 59 59 59 59 59 61
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61 61 62 62 62
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CONTENTS
10.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 10.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 10.8 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 11 Magma
11.1 History and terminology . . 11.2 Definition . . . . . . . . . . 11.3 Morphism of magmas . . . . 11.4 Notation and combinatorics . 11.5 Free magma . . . . . . . . . 11.6 Types of magmas . . . . . . 11.7 Classification by properties . 11.8 Generalizations . . . . . . . 11.9 See also . . . . . . . . . . . 11.10References . . . . . . . . . 11.11Further reading . . . . . . .
64
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12 Order
12.1 Example . . . . . . . . . . . . 12.2 Order and structure . . . . . . 12.3 Counting by order of elements 12.4 In relation to homomorphisms . 12.5 Class equation . . . . . . . . . 12.6 Open questions . . . . . . . . 12.7 References . . . . . . . . . . . 12.8 See also . . . . . . . . . . . .
64 64 64 64 65 65 65 66 66 66 66 67
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13 Ring
13.1 Definition and illustration . . . . . . . . . . . . . . . . 13.1.1 Definition . . . . . . . . . . . . . . . . . . . . 13.1.2 Notes on the definition . . . . . . . . . . . . . 13.1.3 Basic properties . . . . . . . . . . . . . . . . . 13.1.4 Example: Integers modulo 4 . . . . . . . . . . 13.1.5 Example: 2-by-2 matrices . . . . . . . . . . . . 13.2 History . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.1 Dedekind . . . . . . . . . . . . . . . . . . . . 13.2.2 Hilbert . . . . . . . . . . . . . . . . . . . . . 13.2.3 Fraenkel and Noether . . . . . . . . . . . . . . 13.2.4 Multiplicative identity: mandatory vs. optional . 13.3 Basic examples . . . . . . . . . . . . . . . . . . . . . 13.4 Basic concepts . . . . . . . . . . . . . . . . . . . . . 13.4.1 Elements in a ring . . . . . . . . . . . . . . . .
67 67 68 68 68 68 68 68 69
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69 70 70 70 70 70 71 71 71 71 71 72 73 73
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CONTENTS
13.4.2 Subring . . . . . . . . . . . . . . . . . . . . . . 13.4.3 Ideal . . . . . . . . . . . . . . . . . . . . . . . . 13.4.4 Homomorphism . . . . . . . . . . . . . . . . . . 13.4.5 Quotient ring . . . . . . . . . . . . . . . . . . . 13.5 Ring action: a module over a ring . . . . . . . . . . . . . 13.6 Constructions . . . . . . . . . . . . . . . . . . . . . . . 13.6.1 Direct product . . . . . . . . . . . . . . . . . . 13.6.2 Polynomial ring . . . . . . . . . . . . . . . . . . 13.6.3 Matrix ring and endomorphism ring . . . . . . . 13.6.4 Limits and colimits of rings . . . . . . . . . . . 13.6.5 Localization . . . . . . . . . . . . . . . . . . . 13.6.6 Completion . . . . . . . . . . . . . . . . . . . . 13.6.7 Rings with generators and relations . . . . . . . . 13.7 Special kinds of rings . . . . . . . . . . . . . . . . . . . 13.7.1 Domains . . . . . . . . . . . . . . . . . . . . . 13.7.2 Division ring . . . . . . . . . . . . . . . . . . . 13.7.3 Semisimple rings . . . . . . . . . . . . . . . . . 13.7.4 Central simple algebra and Brauer group . . . . . 13.7.5 Valuation ring . . . . . . . . . . . . . . . . . . 13.8 Rings with extra structure . . . . . . . . . . . . . . . . . 13.9 Some examples of the ubiquity of rings . . . . . . . . . . 13.9.1 Cohomology ring of a topological space . . . . . 13.9.2 Burnside ring of a group . . . . . . . . . . . . . 13.9.3 Representation ring of a group ring . . . . . . . . 13.9.4 Function field of an irreducible algebraic variety . 13.9.5 Face ring of a simplicial complex . . . . . . . . . 13.10Category theoretical description . . . . . . . . . . . . . 13.11Generalization . . . . . . . . . . . . . . . . . . . . . . 13.11.1 Rng . . . . . . . . . . . . . . . . . . . . . . . . 13.11.2 Nonassociative ring . . . . . . . . . . . . . . . . 13.11.3 Semiring . . . . . . . . . . . . . . . . . . . . . 13.12Other ring-like objects . . . . . . . . . . . . . . . . . . 13.12.1 Ring object in a category . . . . . . . . . . . . . 13.12.2 Ring scheme . . . . . . . . . . . . . . . . . . . 13.12.3 Ring spectrum . . . . . . . . . . . . . . . . . . 13.13See also . . . . . . . . . . . . . . . . . . . . . . . . . . 13.14Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.15Citations . . . . . . . . . . . . . . . . . . . . . . . . . . 13.16References . . . . . . . . . . . . . . . . . . . . . . . . 13.16.1 General references . . . . . . . . . . . . . . . . 13.16.2 Special references . . . . . . . . . . . . . . . . .
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73 73 74 74 75 75 75 76 76 77 77 78 78 78 78 79 79 79 80 80 81 81 81 81 81 81 81 82 82 82 82 82 82 82 82 82 83 83 84 84 85
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13.16.3 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 13.16.4 Historical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 14 Subgroup
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14.1 Basic properties of subgroups . . . . . . . . . . . . . . . . . . . 14.2 Cosets and Lagrange’s theorem . . . . . . . . . . . . . . . . . . 14.3 Example: Subgroups of Z8 . . . . . . . . . . . . . . . . . . . . 14.4 Example: Subgroups of S₄ (the symmetric group on 4 elements) . 14.4.1 12 elements . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 8 elements . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 6 elements . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 4 elements . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.5 3 elements . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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15 Symmetry
15.1 In mathematics . . . . . . . . . . . 15.1.1 In geometry . . . . . . . . . 15.1.2 In logic . . . . . . . . . . . 15.1.3 Other areas of mathematics . 15.2 In science and nature . . . . . . . . 15.2.1 In physics . . . . . . . . . . 15.2.2 In biology . . . . . . . . . . 15.2.3 In chemistry . . . . . . . . . 15.3 In social interactions . . . . . . . . . 15.4 In the arts . . . . . . . . . . . . . . 15.4.1 In architecture . . . . . . . . 15.4.2 In pottery and metal vessels . 15.4.3 In quilts . . . . . . . . . . . 15.4.4 In carpets and rugs . . . . . 15.4.5 In music . . . . . . . . . . . 15.4.6 In other arts and crafts . . . 15.4.7 In aesthetics . . . . . . . . . 15.4.8 In literature . . . . . . . . . 15.5 See also . . . . . . . . . . . . . . . 15.6 Notes . . . . . . . . . . . . . . . . 15.7 References . . . . . . . . . . . . . . 15.8 Further reading . . . . . . . . . . . 15.9 External links . . . . . . . . . . . .
87 88 88 88 88 88 88 88 88 88 89 89 89 90
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91 91 91 92 92 92 92 93 93 93 93 93 94 94 94 95 95 95 95 95 95 96 97
CONTENTS
ix
16 Symmetry group
98
16.1 Introduction . . . . . . . . . 16.2 One dimension . . . . . . . . 16.3 Two dimensions . . . . . . . 16.4 Three dimensions . . . . . . 16.5 Symmetry groups in general . 16.6 See also . . . . . . . . . . . 16.7 Further reading . . . . . . . 16.8 External links . . . . . . . .
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17 Vector field
17.1 Definition . . . . . . . . . . . . . . . . . . . . . . 17.1.1 Vector fields on subsets of Euclidean space . 17.1.2 Coordinate transformation law . . . . . . . 17.1.3 Vector fields on manifolds . . . . . . . . . . 17.2 Examples . . . . . . . . . . . . . . . . . . . . . . 17.2.1 Gradient field . . . . . . . . . . . . . . . . 17.2.2 Central field . . . . . . . . . . . . . . . . . 17.3 Operations on vector fields . . . . . . . . . . . . . 17.3.1 Line integral . . . . . . . . . . . . . . . . 17.3.2 Divergence . . . . . . . . . . . . . . . . . 17.3.3 Curl . . . . . . . . . . . . . . . . . . . . . 17.3.4 Index of a vector field . . . . . . . . . . . . 17.4 History . . . . . . . . . . . . . . . . . . . . . . . . 17.5 Flow curves . . . . . . . . . . . . . . . . . . . . . 17.5.1 Complete vector fields . . . . . . . . . . . 17.6 Difference between scalar and vector field . . . . . . 17.6.1 Example 1 . . . . . . . . . . . . . . . . . . 17.6.2 Example 2 . . . . . . . . . . . . . . . . . . 17.7 f-relatedness . . . . . . . . . . . . . . . . . . . . . 17.8 Generalizations . . . . . . . . . . . . . . . . . . . 17.9 See also . . . . . . . . . . . . . . . . . . . . . . . 17.10References . . . . . . . . . . . . . . . . . . . . . . 17.11Bibliography . . . . . . . . . . . . . . . . . . . . . 17.12External links . . . . . . . . . . . . . . . . . . . .
98 99 99 100 100 101 101 101 102
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18 Vector space
18.1 Introduction and definition . . . . . . . . . . . . . . . . . . . 18.1.1 First example: arrows in the plane . . . . . . . . . . . 18.1.2 Second example: ordered pairs of numbers . . . . . . . 18.1.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . 18.1.4 Alternative formulations and elementary consequences .
102 102 103 103 103 104 104 104 104 105 105 105 105 106 106 106 106 106 107 107 107 107 107 107 108
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x
CONTENTS
18.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . 18.3.1 Coordinate spaces . . . . . . . . . . . . . . . . 18.3.2 Complex numbers and other field extensions . . 18.3.3 Function spaces . . . . . . . . . . . . . . . . . 18.3.4 Linear equations . . . . . . . . . . . . . . . . 18.4 Basis and dimension . . . . . . . . . . . . . . . . . . 18.5 Linear maps and matrices . . . . . . . . . . . . . . . . 18.5.1 Matrices . . . . . . . . . . . . . . . . . . . . . 18.5.2 Eigenvalues and eigenvectors . . . . . . . . . . 18.6 Basic constructions . . . . . . . . . . . . . . . . . . . 18.6.1 Subspaces and quotient spaces . . . . . . . . . 18.6.2 Direct product and direct sum . . . . . . . . . 18.6.3 Tensor product . . . . . . . . . . . . . . . . . 18.7 Vector spaces with additional structure . . . . . . . . . 18.7.1 Normed vector spaces and inner product spaces 18.7.2 Topological vector spaces . . . . . . . . . . . . 18.7.3 Algebras over fields . . . . . . . . . . . . . . . 18.8 Applications . . . . . . . . . . . . . . . . . . . . . . . 18.8.1 Distributions . . . . . . . . . . . . . . . . . . 18.8.2 Fourier analysis . . . . . . . . . . . . . . . . . 18.8.3 Differential geometry . . . . . . . . . . . . . . 18.9 Generalizations . . . . . . . . . . . . . . . . . . . . . 18.9.1 Vector bundles . . . . . . . . . . . . . . . . . 18.9.2 Modules . . . . . . . . . . . . . . . . . . . . . 18.9.3 Affine and projective spaces . . . . . . . . . . . 18.10See also . . . . . . . . . . . . . . . . . . . . . . . . . 18.11Notes . . . . . . . . . . . . . . . . . . . . . . . . . . 18.12Footnotes . . . . . . . . . . . . . . . . . . . . . . . . 18.13References . . . . . . . . . . . . . . . . . . . . . . . . 18.13.1 Algebra . . . . . . . . . . . . . . . . . . . . . 18.13.2 Analysis . . . . . . . . . . . . . . . . . . . . . 18.13.3 Historical references . . . . . . . . . . . . . . 18.13.4 Further references . . . . . . . . . . . . . . . . 18.14External links . . . . . . . . . . . . . . . . . . . . . .
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19 Zero divisor
19.1 Examples . . . . . . . . . . . 19.1.1 One-sided zero-divisor 19.2 Non-examples . . . . . . . . . 19.3 Properties . . . . . . . . . . . 19.4 Zero as a zero divisor . . . . .
110 110 110 110 110 111 111 112 113 113 113 114 114 114 115 115 116 117 118 118 118 119 119 120 120 120 121 121 121 123 123 123 124 124 125 126
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xi
CONTENTS
19.5 Zero divisor on a module . . . . . . . . . . . . . 19.6 See also . . . . . . . . . . . . . . . . . . . . . . 19.7 Notes . . . . . . . . . . . . . . . . . . . . . . . 19.8 References . . . . . . . . . . . . . . . . . . . . 19.9 Text and image sources, contributors, and licenses 19.9.1 Text . . . . . . . . . . . . . . . . . . . . 19.9.2 Images . . . . . . . . . . . . . . . . . . 19.9.3 Content license . . . . . . . . . . . . . .
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127 127 127 127 128 128 132 136
Chapter 1
Abelian group For the group described by the archaic use of the related More compactly, an abelian group is a commutative term “Abelian linear group”, see Symplectic group. group. A group in which the group operation is not commutative is called a “non-abelian group” or “nonIn abstract algebra, an abelian group, also called a com- commutative group”. mutative group , is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the 1.2 Facts axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers. They are named after 1.2.1 Notation Niels Henrik Abel.[1] The concept of an abelian group is one of the first See also: Additive group and Multiplicative group concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules There are two main notational conventions for abelian and vector spaces are developed. The theory of abelian groups – additive and multiplicative. groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well un- Generally, the multiplicative notation is the usual notaderstood. On theother hand, thetheoryof infinite abelian tion for groups, while the additive notation is the usual notation for modules and rings. The additive notation groups is an area of current research. may also be used to emphasize that a particular group is abelian, whenever both abelian and non-abelian groups are considered, some notable exceptions being near-rings 1.1 Definition and partially ordered groups, where an operation is written additively even when non-abelian. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b. The symbol • is a general place- 1.2.2 Multiplication table holder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy To verify that a finite group is abelian, a table (matrix) – known as a Cayley table – can be constructed in a similar five requirements known as the abelian group axioms : fashion to a multiplication table. If the group is G = { g1 = e , g 2 , ..., gn} under the operation ⋅, the ( i , j )th entry Closure For all a, b in A, the result of the operation a • of this table contains the product gi ⋅ gj . The group is b is also in A. abelian if and only if this table is symmetric about the main diagonal. Associativity For all a, b and c in A, the equation (a • b) • c = a • (b • c ) holds. This is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi . This implies that the ( i , j )th entry of the table equals Identity element There exists an element e in A , such the ( j , i )th entry, thus the table is symmetric about the that for all elements a in A, the equation e • a = a • main diagonal. e = a holds. Inverse element For each a in A, there exists an element
1.3
b in A such that a • b = b • a = e , where e is the
identity element.
Examples
• For the integers and the operation addition "+", de-
Commutativity For all a, b in A, a • b = b • a.
noted (Z, +), the operation + combines any two inte-
1
2
CHAPTER 1. ABELIAN GROUP
gers to form a third integer, addition is associative, zero is the additive identity, every integer n has an additive inverse, −n, and the addition operation is commutative since m + n = n + m for any two integers m and n.
• Every cyclic groupm G n is abelian, because if x , y are m + n n + m n m
in G , then xy = a a = a = a = a a = yx . Thus the integers, Z , form an abelian group under addition, as do the integers modulo n, Z/nZ.
• Every ring is an abelian group with respect to
its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers arean abelian group under addition, and thenonzero real numbers are an abelian group under multiplication.
• Every subgroup of an abelian group is normal, so
each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime order. [2]
• The concepts of abelian group and Z-module agree.
More specifically, every Z -module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers Z in a unique way.
Theorems about abelian groups (i.e. modules over the principal ideal domain Z) can often be generalized to theorems about modules over an arbitrary principal ideal domain. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain. In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Z / pk Z for p prime, and the latter is a direct sum of finitely many copies of Z. If f , g : G → H are two group homomorphisms between abelian groups, then their sum f + g , defined by ( f + g ) ( x ) = f ( x ) + g( x ), is again a homomorphism. (This is not true if H is a non-abelian group.) The set Hom( G , H ) of all group homomorphisms from G to H thus turns into an abelian group in its own right. Somewhat akin to the dimension of vector spaces, every abelian group has a rank . It is defined as the cardinality of the largest set of linearly independent elements of the group. The integers and the rational numbers have rank one, as well as every subgroup of the rationals. The center Z (G ) of a group G is the set of elements that commute with every element of G . A group G is abelian if and only if it is equal to its center Z (G ). The center of a group G is always a characteristic abelian subgroup of G . If the quotient group G /Z (G ) of a group by its center is cyclic then G is abelian.[3]
In general, matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative. How- 1.6 Finite abelian groups ever, some groups of matrices are abelian groups under matrix multiplication – one example is the group of 2×2 Cyclic groups of integers modulo n , Z /nZ, were among rotation matrices. the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of 1.4 Historical remarks invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariAbelian groups were named after Norwegian ants. The theory had been first developed in the 1879 mathematician Niels Henrik Abel by Camille Jor- paper of Georg Frobenius and Ludwig Stickelberger and dan because Abel found that the commutativity of the later was both simplified and generalized to finitely gengroup of a polynomial implies that the roots of the erated modules over a principal ideal domain, forming an polynomial can be calculated by using radicals. See important chapter of linear algebra. Section 6.5 of Cox (2004) for more information on the Any group of prime order is isomorphic to a cyclic group historical background. and therefore abelian. Any group whose order is a square of a prime number is abelian. [4] In fact, for every prime number p there are (up to isomorphism) exactly two 1.5 Properties groups of order p2, namely Z p2 and Z p×Z p. If n is a natural number and x is an element of an abelian group G written additively, then nx can be defined as x + x 1.6.1 Classification + ... + x (n summands) and (−n) x = −(nx ). In this way, G becomes a module over the ring Z of integers. In fact, the The fundamental theorem of finite abelian groups modules over Z can be identified with the abelian groups. states that every finite abelian group G can be expressed
3
1.7. INFINITE ABELIAN GROUPS
as the direct sum of cyclic subgroups of prime-power order. This is a special case of the fundamental theorem of finitely generated abelian groups when G has zero rank. The cyclic group Zmn oforder mn is isomorphic to thedirectsumof Zm and Zn ifandonlyif m and n are coprime. It follows that any finite abelian group G is isomorphic to a direct sum of the form u
⊕
Zki
i=1
group can be used. Another special case is when n is arbitrary but ei = 1 for 1 ≤ i ≤ n. Here, one is considering P to be of the form Z p
⊕···⊕Z , p
so elements of this subgroup can be viewed as comprising a vector space of dimension n over the finite field of p elements F p. The automorphisms of this subgroup are therefore given by the invertible linear transformations, so
in either of the following canonical ways:
• the numbers k 1, ..., ku are powers of primes • k 1 divides k 2, which divides k 3, and so on up to ku. For example, Z 15 can be expressed as the direct sum of two cyclic subgroups of order 3 and 5: Z15 ≅ {0, 5, 10} ⊕ {0, 3, 6, 9, 12}. The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. For another example, every abelian group of order 8 is isomorphic to either Z 8 (the integers 0 to 7 under addition modulo 8), Z4 ⊕ Z2 (the odd integers 1 to 15 under multiplication modulo 16), or Z2 ⊕ Z2 ⊕ Z2. See also list of small groups for finite abelian groups of order 16 or less. 1.6.2
Automorphisms
One can apply the fundamental theorem to count (and sometimes determine) the automorphisms of a given finite abelian group G . To do this, one uses the fact that if G splits as a direct sum H ⊕ K of subgroups of coprime order, then Aut(H ⊕ K ) ≅ Aut(H ) ⊕ Aut(K ). Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p-subgroups separately (that is, all direct sums of cyclic subgroups, each with order a power of p). Fix a prime p and suppose the exponents ei of the cyclic factors of theSylow p-subgroup are arranged in increasing order:
≤ e2 ≤ · · · ≤ e
e1
n
for some n > 0. One needs to find the automorphisms of Z pe1
⊕ · · · ⊕ Z
pen .
One special case is when n = 1, so that there is only one cyclic prime-power factor in the Sylow p-subgroup P . In this case the theory of automorphisms of a finite cyclic
Aut(P ) ∼ = GL(n, F p ), where GL is the appropriate general linear group. This is easily shown to have order
|Aut(P )| = ( p − 1) ··· ( p − p −1). n
n
n
In the most general case, where the ei and n are arbitrary, the automorphism group is more difficult to determine. It is known, however, that if one defines dk = max r er = e k
{|
}
and ck = min r er = e k
{|
}
then one has in particular dk ≥ k , ck ≤ k , and n
|Aut(P )| =
�
k =1
n
dk
( p
k 1
− p
−)
�
n
ej n d j
( p
) −
j =1
�
( pei −1 )n−ci +1 .
i=1
One can check that this yields the orders in the previous examples as special cases (see [Hillar,Rhea]).
1.7
Infinite abelian groups
Тhe simplest infinite abelian group is the infinite cyclic group Z . Any finitely generated abelian group A is isomorphic to the direct sum of r copies of Z and a finite abelian group, which in turn is decomposable into a direct sum of finitely many cyclic groups of primary orders. Even though the decomposition is not unique, the number r , called the rank of A, and the prime powers giving the orders of finite cyclic summands are uniquely determined. By contrast, classification of general infinitely generated abelian groupsis far fromcomplete. Divisible groups, i.e.
4
CHAPTER 1. ABELIAN GROUP
abeliangroups A in which theequation nx = a admitsasolution x ∈ A for any natural number n and element a of A, constitute one important class of infinite abelian groups that can be completely characterized. Every divisible group is isomorphic to a direct sum, with summands isomorphic to Q and Prüfer groups Q p/Z p for various prime numbers p, and the cardinality of the set of summands of each type is uniquely determined. [5] Moreover, if a divisible group A is a subgroup of an abelian group G then A admits a direct complement: a subgroup C of G such that G = A ⊕ C . Thus divisible groups are injective modules in the category of abelian groups, and conversely, every injective abelian group is divisible (Baer’s criterion). An abelian group without non-zero divisible subgroups is called reduced. Two important special classes of infinite abelian groups with diametrically opposite properties are torsion groups and torsion-free groups , exemplified by the groups Q /Z (periodic) and Q (torsion-free). 1.7.1
Torsion groups
An abelian group is called periodic or torsion if every element has finite order. A direct sum of finite cyclic groups is periodic. Although the converse statement is not true in general, some special cases are known. The first and second Prüfer theorems state that if A is a periodic group and either it has bounded exponent, i.e. nA = 0 for some natural number n, or if A is countable and the p-heights of the elements of A are finite for each p, then A is isomorphic to a direct sum of finite cyclic groups. [6] The cardinality of the set of direct summands isomorphic to Z/ pm Z in such a decomposition is an invariant of A. These theorems were later subsumed in the Kulikov criterion. In a different direction, Helmut Ulm found an extension of the second Prüfer theorem to countable abelian p -groups with elements of infinite height: those groups are completely classified by means of their Ulm invariants. 1.7.2
Torsion-free and mixed groups
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively:
• Free abelian groups, i.e. arbitrary direct sums of Z • Cotorsion and algebraically compact torsion-free groups such as the p-adic integers
• Slender groups An abelian group that is neither periodic nor torsionfree is called mixed. If A is an abelian group and T (A) is its torsion subgroup then the factor group A/T (A) is torsion-free. However, in general the torsion subgroup
is not a direct summand of A, so A is not isomorphic to T (A) ⊕ A/T (A). Thus thetheoryof mixed groupsinvolves more than simply combining the results about periodic and torsion-free groups. 1.7.3
Invariants and classification
One of the most basic invariants of an infinite abelian group A isits rank: thecardinalityof themaximal linearly independent subset of A . Abelian groups of rank 0 are precisely the periodic groups, while torsion-free abelian groups of rank 1 are necessarily subgroups of Q and can be completely described. More generally, a torsion-free abelian group of finite rank r is a subgroup of Qr . On the other hand, the group of p -adic integers Z p is a torsionfree abelian group of infinite Z-rank and the groups Z pn with different n are non-isomorphic, so this invariant does not even fully capture properties of some familiar groups. The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before 1950 and form a foundation of the classification of more general infinite abelian groups. Important technical tools used in classification of infinite abelian groups are pure and basic subgroups. Introduction of various invariants of torsion-free abelian groupshasbeen oneavenue of further progress. See the books by Irving Kaplansky, László Fuchs, Phillip Griffith, and David Arnold, as well as the proceedings of the conferences on Abelian Group Theory published in Lecture Notes in Mathematics for more recent results. 1.7.4
Additive groups of rings
The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication). Some important topics in this area of study are:
• Tensor product • Corner’s results on countable torsion-free groups • Shelah’s work to remove cardinality restrictions. 1.8
Relation to other mathematical topics
Many large abelian groups possess a natural topology, which turns them into topological groups. The collection of all abelian groups, together with the homomorphisms between them, forms the category Ab, the prototype of an abelian category.
5
1.10. SEE ALSO
Nearly all well-known algebraic structures other than 1.10 See also Boolean algebras are undecidable. Hence it is surprising that Tarski’s student Wanda Szmielew (1955) proved • Abelianization that the first order theory of abelian groups, unlike its • Class field theory nonabelian counterpart, is decidable. This decidability, plus the fundamental theorem of finite abelian groups de• Commutator subgroup scribed above, highlight some of the successes in abelian group theory, but there are still many areas of current re• Dihedral group of order 6, the smallest non-Abelian search: group
• Amongst torsion-free abelian groups of finite rank,
only the finitely generated case and the rank 1 case are well understood;
• There are many unsolved problems in the theory of infinite-rank torsion-free abelian groups;
• Elementary abelian group • Pontryagin duality • Pure injective module • Pure projective module
• While countable torsion abelian groups are well un-
derstood through simple presentations and Ulm in- 1.11 Notes variants, thecase of countable mixed groupsis much [1] Jacobson (2009), p. 41 less mature.
• Many mild extensions of the first order theory of abelian groups are known to be undecidable.
• Finite abelian groups remain a topic of research in computational group theory.
Moreover, abelian groups of infinite order lead, quite surprisingly, to deep questions about the set theory commonly assumed to underlie all of mathematics. Take the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem is:
[2] Rose 2012, p. 32 [3] Rose 2012, p. 48 [4] Rose 2012, p. 79 [5] For example, Q/Z ≅ ∑ p Q p/Z p. [6] Countability assumption in the second Prüfer theorem cannot be removed: the torsion subgroup of the direct product of the cyclic groups Z/ pm Z for all natural m is not a direct sum of cyclic groups. [7] “Abel Prize Awarded: The Mathematicians’ Nobel”. Archived from the original on 1 July 2013. Retrieved 3 July 2016.
• Undecidable in ZFC (Zermelo–Fraenkel axioms),
•
the conventional axiomatic set theory from which 1.12 References nearly all of present-day mathematics can be derived. The Whitehead problem is also the first ques• Cox, David (2004). Galois Theory. Wileytion in ordinary mathematics proved undecidable in Interscience. MR 2119052. ZFC; • Fuchs, László (1970). Infinite Abelian Groups. Pure and Applied Mathematics 36–I. Academic Press. Undecidable even if ZFC is augmented bytakingthe MR 0255673. generalized continuum hypothesis as an axiom;
• Positively answered if ZFC is augmented with the
axiom of constructibility (see statements true in L).
1.9
A note on the typography
Among mathematical adjectives derived from the proper name of a mathematician, the word “abelian” is rare in that it is often spelled with a lowercase a, rather than an uppercase A, indicating how ubiquitous the concept is in modern mathematics.[7]
• Fuchs, László (1973). Infinite Abelian Groups. Pure
and Applied Mathematics. 36-II. Academic Press. MR 0349869.
• Griffith,
Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics.
University of Chicago Press. ISBN 0-226-30870-7.
• Rose, John S. (2012).
A Course on Group The-
ory. Dover Publications. ISBN 0-486-68194-7.
Unabridged and unaltered republication of a work first published by the Cambridge University Press, Cambridge, England, in 1978.
6
CHAPTER 1. ABELIAN GROUP
• Herstein, I. N. (1975). Topics in Algebra (2nd ed.). John Wiley & Sons. ISBN 0-471-02371-X.
• Hillar, Christopher;
Rhea, Darren (2007). “Automorphisms of finite abelian groups”. American Mathematical Monthly 114 (10): 917–923. arXiv:math/0605185.
• Jacobson, Nathan (2009). Basic Algebra I (2nd ed.). Dover Publications. ISBN 978-0-486-47189-1.
• Szmielew, Wanda (1955).
“Elementary properties of abelian groups”. Fundamenta Mathematicae 41 : 203–271.
1.13
External links
• Hazewinkel, Michiel, ed. (2001), “Abelian group”, Encyclopedia of Mathematics , Springer, ISBN 978-
1-55608-010-4
Chapter 2
Category theory study of monads in functional programming.
2.1 Basic concepts Categories represent abstraction of other mathematical concepts. Many areas of mathematics can be formalised by category theory as categories. Hence category theory uses abstraction to make it possible to state and prove many intricate and subtle mathematical results in these fields in a much simpler way. [2] A basic example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another. However, the objects of a category need not be sets, and the arrows need not be functions. Any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The “arrows” of category theory are often said to represent a process connecting two objects, or in many cases a “structure-preserving” transformation connecting two objects. There are, however, many applications where much more abstract concepts are represented by objects and morphisms. The most important property of the arrows is that they can be “composed”, in other words, arranged in a sequence to form a new arrow.
Schematic representation of a category with objects X , Y , Z and morphisms f , g , g ∘ f. (The category’s three identity morphisms 1X , 1 Y and 1Z , if explicitly represented, would appear as three arrows, next to the letters X, Y, and Z, respectively, each having as its “shaft” a circular arc measuring almost 360 degrees.) Category theory[1]
formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category hastwo basic properties: the abilityto compose thearrows associatively and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Several terms used in category theory, including the term “morphism”, are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself. Samuel Eilenberg and SaundersMac Lane introduced the concepts of categories, functors, and natural transformations in 1942–45 in their study of algebraic topology, with thegoal of understanding theprocesses that preserve mathematical structure. Category theory has practical applications in programming language theory, in particular for the
2.2
Applications of Categories
Categories now appear in many branches of mathematics, some areas of theoretical computer science where they can correspond to types, and mathematical physics where they can be used to describe vector spaces.[3] Linear algebra can also be expressed in terms of categories of matrices. [4]
2.3 7
Utility
8 2.3.1
CHAPTER 2. CATEGORY THEORY
Categories, objects, and morphisms
Thestudyof categories isanattemptto axiomatically capture what is commonly found in various classes of related mathematical structures by relating them to the structure preserving functions between them. A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures from the axioms of a category. Consider the following example. The class Grp of groups consists of all objects having a “group structure”. One can proceed to prove theorems about groups by making logical deductions from the set of axioms. For example, it is immediately proven from the axioms that the identity element of a group is unique. Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, we are able to learn more about the structure of the objects. In the case of groups, the morphisms are the group homomorphisms. A group homomorphism betweentwo groups“preserves thegroup structure” in a precise sense – it is a “process” taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms. A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps (morphisms) between topological spaces in topology (the associated category is called Top), and the study of smooth functions (morphisms) in manifold theory. Not all categories arise as “structure preserving (set) functions”, however; the standard example is the category of homotopies between pointed topological spaces. If one axiomatizes relations instead of functions, one obtains the theory of allegories. 2.3.2
Functors
Main article: Functor See also: Adjoint functors § Motivation
one category an object of another category, and to every morphism in the first category a morphism in the second. In fact, what we have done is define a category of cate gories and functors – the objects are categories, and the morphisms (between categories) are functors. By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them; we are studying the relationships between various classes of mathematical structures . This is a fundamental idea, which first surfaced in algebraic topology. Difficult topological questions can be translated into algebraic questions which are often easier to solve. Basic constructions, such as the fundamental group or the fundamental groupoid of a topological space, can be expressed as functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications. 2.3.3
Natural transformations
Main article: Natural transformation Abstracting yetagain, somediagrammatic and/or sequential constructions are often “naturally related” – a vague notion, at first sight. This leads to the clarifying concept of natural transformation, a way to “map” one functor to another. Many important constructions in mathematics can be studied in this context. “Naturality” is a principle, like general covariance in physics, that cuts deeper than is initially apparent. An arrow between two functors is a natural transformation when it is subject to certain naturality or commutativity conditions. Functors and natural transformations ('naturality') are the key concepts in category theory. [5]
2.4 Categories, objects, and morphisms Main articles: Category (mathematics) and Morphism
2.4.1
Categories
A category C consists of the following threemathematical A category is itself a type of mathematical structure, so entities: we can look for “processes” which preserve this structure in some sense; such a process is called a functor. • A class ob(C ), whose elements are called objects; Diagram chasing is a visual method of arguing with ab• A class hom(C ), whose elements are called stract “arrows” joined in diagrams. Functors are represented by arrows between categories, subject to spemorphisms or maps or arrows. Each morphism f cific defining commutativity conditions. Functors can dehas a source object a and target object b. fine (construct) categorical diagrams and sequences (viz. The expression f : a → b, would be verbally stated Mitchell, 1965). A functor associates to every object of as " f is a morphism from a to b".
9
2.5. FUNCTORS
The expression hom(a, b) — alternatively expressed as homC (a, b), mor(a, b), or C (a, b) — denotes the hom-class of all morphisms from a to b.
• A binary operation ∘, called composition of mor-
phisms, such that for any three objects a , b , and c , we have hom(b, c ) × hom(a, b ) → hom(a, c ). The composition of f : a → b and g : b → c is written as g ∘ f or gf ,[6] governed by two axioms:
• Associativity: If f : a → b, g : b → c and h : c → d then h ∘ ( g ∘ f ) = (h ∘ g) ∘ f , and • Identity: For every object x , there exists a morphism 1 x : x → x called the identity morphism for x , such that for every morphism f : a → b, we have 1b ∘ f = f = f ∘ 1 a. From the axioms, it can be proved that there is exactly one identity morphism for every object. Some authors deviate from the definition just given by identifying each object with its identity morphism. 2.4.2
Morphisms
Relations among morphisms (such as fg = h) are often depicted using commutative diagrams, with “points” (corners) representing objects and “arrows” representing morphisms. Morphisms can have any of the following properties. A morphism f : a → b is a:
• f is a monomorphism and a retraction; • f is an epimorphism and a section; • f is an isomorphism. 2.5
Functors
Main article: Functor Functors are structure-preserving maps between categories. They can be thought of as morphisms in the category of all (small) categories. A (covariant) functor F from a category C to a category D , written F : C → D , consists of:
• for each object x in C , an object F ( x ) in D ; and • for each morphism f : x → y in C , a morphism F ( f ) : F ( x ) → F (y),
such that the following two properties hold:
• For every object x in C , F (1 x ) = 1F ₍ x ₎; • For all morphisms f : x → y and g : y → z, F ( g ∘ f ) = F ( g) ∘ F ( f ).
A contravariant functor F : C → D , is like a covariant functor, except that it “turns morphisms around” (“reverses all the arrows”). More specifically, every morphism f : x → y in C must be assigned to a morphism F ( f ) : F (y) → F ( x ) in D . In other words, a contravariant functor acts as a covariant functor from the opposite • monomorphism (or monic ) if f ∘ g1 = f ∘ g2 implies category C op to D . g1 = g2 for all morphisms g1 , g2 : x → a.
• epimorphism (or epic ) if g1 ∘ f = g2 ∘ f implies g1
2.6
• bimorphism if f is both epic and monic. • isomorphism if there exists a morphism g : b → a [7]
Main article: Natural transformation
= g2 for all morphisms g1, g2 : b → x .
such that f ∘ g = 1b and g ∘ f = 1a.
• endomorphism if a = b. end(a) denotes the class of endomorphisms of a.
• automorphism if f is both an endomorphism and an isomorphism. aut(a) denotes the class of automorphisms of a.
• retraction if a right inverse of f exists, i.e. if there exists a morphism g : b → a with f ∘ g = 1b.
• section if a left inverse of f exists, i.e. if there exists a morphism g : b → a with g ∘ f = 1a.
Every retraction is an epimorphism, andevery section is a monomorphism. Furthermore, the following three statements are equivalent:
Natural transformations
A natural transformation is a relation between two functors. Functors often describe “natural constructions” and natural transformations then describe “natural homomorphisms” between two such constructions. Sometimes two quite different constructions yield “the same” result; this is expressed by a natural isomorphism between the two functors. If F and G are (covariant) functors between the categories C and D , then a natural transformation η from F to G associates to every object X in C a morphism ηX : F (X ) → G (X ) in D such that for every morphism f : X → Y in C , we have η Y ∘ F ( f ) = G ( f ) ∘ ηX ; this means that the following diagram is commutative: The two functors F and G are called naturally isomorphic if there exists a natural transformation from F to G such that ηX is an isomorphism for every object X in C .
10
CHAPTER 2. CATEGORY THEORY
can two categories be considered essentially the same , in the sense that theorems about one category can readily be transformed into theorems about the other category? The major tool one employs to describe such a situation is called equivalence of categories , which is given by appropriate functors between two categories. Categorical equivalence has found numerous applications in mathematics. 2.7.3
Commutative diagram defining natural transformations
2.7 Other concepts 2.7.1
Universal constructions, limits, and colimits
Main articles: Universal property and Limit (category theory) Using the language of category theory, many areas of mathematical study can be categorized. Categories include sets, groups and topologies. Each category is distinguished by properties that all its objects have in common, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered atomic, i.e., we do not know whether an object A is a set, a topology, or any other abstract concept. Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest. Indeed, it turns out that numerous important constructions can be described in a purely categorical way. The central concept which is needed for this purpose is called categorical limit , and can be dualized to yield the notion of a colimit .
Further concepts and results
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
• The functor category D C has as objects the functors
from C to D and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes representable functors in functor categories.
• Duality: Every statement, theorem, or definition in
category theory has a dual which is essentially obtainedby“reversing allthearrows”. If onestatement is true in a category C then its dual is true in the dual category C op. This duality, which is transparent at the level of category theory, is often obscured in applications and can lead to surprising relationships.
• Adjoint functors:
A functor can be left (or right) adjoint to another functor that maps in the opposite direction. Such a pair of adjoint functors typically arises from a construction defined by a universal property; this can be seen as a more abstract and powerful view on universal properties.
2.7.4
Higher-dimensional categories
Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories . Briefly, if we consider a morphism between two objects as a “process taking us from one object to another”, then higher-dimensional categories allow us to profitably generalize this by considering “higherdimensional processes”. For example, a (strict) 2-category is a category together with “morphisms between morphisms”, i.e., processes 2.7.2 Equivalent categories which allow us to transform one morphism into another. We can then “compose” these “bimorphisms” both horMain articles: Equivalence of categories and izontally and vertically, and we require a 2-dimensional Isomorphism of categories “exchange law” to hold, relating the two composition laws. In this context, the standard example is Cat , the It is a natural question to ask: under which conditions 2-category of all (small) categories, and in this example,
2.9. SEE SEE ALSO
bimorphisms of morphisms are simply natural simply natural transformations of mations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal essentially monoidal categories. categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative “up to” an isomorphism. This process can be extended for all natural numbers n , and these are called n-categories -categories.. There is even even a notion of ω-category corresponding corresponding to the ordinal the ordinal number ω. ω. Higher-dimensional categories are part of the broader mathematical field of higher-dimensional of higher-dimensional algebra, algebra , a concept introduced by Ronald by Ronald Brown. Brown. For a conversational introduction introduction to these ideas, see John see John Baez, 'A Tale of ncategories’ categori es’ (1996).
2.8
Histo Histori rical cal notes notes
11 as a specific type of category with two additional topos axioms. These foundational applications of category theory have been worked out in fair detail as a basis for, and justification justification of, constructive of, constructive mathematics. mathematics. Topos theory is a form of abstract sheaf abstract sheaf theory, theory, with geometric origins, and leads to ideas such as pointless as pointless topology. topology. Categorical logic is logic is now a well-defined field based on type theory for theory for intuitionistic intuitionistic logics, logics, with applications in functional programming and and domain theory, theory, where a cartesian closed category is category is taken as a non-syntactic description of a lambda a lambda calculus. calculus. At the very least, least, category theoretic language clarifies what exactly these related areas have in common (in some abstract some abstract sense). sense). Category theory has been applied in other fields as well. For example, John example, John Baez has Baez has shown a link between Feynmann diagrams in Physic Feynma Physicss and monoidal monoidal categori categories. es.[8] Another application of category theory, more specifically: topos t opos theory, has been made in mathematical music theory, see for example the book The Topos of Music, Geometric Logic of Concepts, Theory, and Performance
by Guerino Mazzola. Mazzola. Main article: Timeline of category theory and related by Guerino mathematics More recent efforts to introduce undergraduates to categories as a foundation for mathematics include those of William Lawvere and Lawvere and Rosebrugh (2003) and LawIn 1942–45, Samuel 1942–45, Samuel Eilenberg and Eilenberg and Saunders Saunders Mac Lane of William and Stephen Schanuel (1997) Schanuel (1997) and Mirroslav Yotov introduc introduced ed categori categories, es, functo functors, rs, and natural natural transf transfor- vere and Stephen mations as part of their work in topology, especially (2012). algebraic algebra ic topology topology.. Their Their work work was an important important part part of the transition from from intuitive and geometric homology geometric homology to axiomatic to axiomatic homology theory. theory . Eilenberg Eilenberg and Mac Lane 2.9 2.9 See also also later later wrote wrote that that their their goal goal was was to und unders erstan tandd natura naturall transtransformations. formations. That required defining defining functors, functors, which re• Domain theory quired categories. Stanislaw Ulam, Ulam, and some writing on his behalf, have • Enriched category theory claimed that related ideas were current in the late 1930s in Poland Poland.. Eilenbe Eilenberg rg was Polis Polish, h, and studied studied mathema mathematic ticss • Glossary of category category theory in Poland in the 1930s. Category theory is also, in some • Group theory sense, a continuation of the work of Emmy of Emmy Noether (one Noether (one of Mac Lane’s Lane’s teachers teachers)) in formali formalizing zingabstr abstract actproc processe esses; s; • Higher category theory Noether realized that understanding a type of mathematical structure requires understanding the processes that • Higher-dimensional algebra preserve preserve that structure. To achieve achieve this understanding, Eilenberg Eilenberg and Mac Lane proposed an axiomatic formalformal• Important publications publications in category theory ization of the relation between structures and the processes that preserve them. • Lambda calculus The subsequent development of category theory was • Outline of category theory theory powered first by the computational needs of homological of homological algebra,, and later by the axiomatic needs of algebraic algebra of algebraic ge• Timelin Timelinee of category theory and related mathematometry,, the field most resistant to being grounded in eiometry ics ther axiomatic ther axiomatic set theory or theory or the Russell-Whitehead the Russell-Whitehead view view of united united founda foundation tions. s. General General category category theory, theory, an extenextension sion of universa universall algeb algebra ra having having many many new featur features es allowallowing for semantic for semantic flexibility flexibility and higher-order and higher-order logic, logic, came 2.1 2.10 Notes tes later; it is now applied throughout mathematics. Certain categories called topoi called topoi (singular (singular topos) can even [1] Awodey, Steve (2010) Steve (2010) [2006]. Category Theory. OxOxserve as an alternative to axiomatic to axiomatic set theory as theory as a founford Logic Guides 49 (2nd ed.). Oxford University Press. dation of mathematics. A topos can also be considered ISBN 978-0-19-923718-0. 978-0-19-923718-0.
12
CHAPTER 2. CATEGORY CATEGORY THEORY THEORY
[2] Geroch, Robert (1985). Mathematical physics ([Repr.] ed.). Chicago: University of Chicago Press. p. 7. ISBN 0-226-28862-5.. Note that theorem 3 is actually easier for 0-226-28862-5 for categories categories in general than it is for the special case of sets. This phenomenon is by no means rare.
• Goldblatt, Robert (2006) [1979]. Topoi: The Cat-
[3] B. Coeck Coecke, e, editor editorNew NewStruc Structure turess fo forr Phys Physics icsNumber831 Number831 in Lecture Notes in Physics. Springer-Verlag, Springer-Verlag, 2011
• Hatcher, William S. (1982). “Ch. 8”. The logical
[4] Maced Macedo, o, H.D.; H.D.; Olive Oliveir ira, a, J.N. J.N. (2013). (2013). “Typi “Typing ng linear algebra: algebra: A biproduct biproduct-ori -oriented ented approach approach”. ”. Science of Computer Programming 78 (11): 2160–2 2160–2191. 191. doi::10.1016/j.scico.2012.07.012 doi 10.1016/j.scico.2012.07.012.. [5] Mac Lane 1998, 1998, p. 18: “As Eilenberg-Mac Eilenberg-Mac Lane first first observed, 'category' has been defined in order to be able to define 'functor' 'functor' and 'functor' has been defined in order to be able to define 'natural transformation'.” fg or [6] Some Some author authorss compos composee in the opp opposi osite te order order,, writi writing ng fg f ∘∘ g for g ∘ f . Computer scientists using category category theory very commonly write f ;; g for g ∘ f
[7] Note that a morphism that is both epic and monic is not necessarily an isomorphism! isomorphism! An elementary counterexcounterexample: in the category consisting of two objects objects A and B, the identity morphisms, and a single morphism f from from A to B, f is is both epic and monic but is not an isomorphism. [8] Baez, J.C.; Stay, M. (2009). “Physic “Physics, s, topology, logic and computation: A Rosetta stone” (PDF). stone” (PDF). arXiv arXiv::0903.0340 0903.0340..
egorial Analysis of Logic . Studies Studies in logic logic and the the foundations of mathematics 94 (Reprint, revised ed.). Dover Publication Publications. s. ISBN ISBN 978-0-486-450261.
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• Herrlich, Horst; Strecker, George E. (2007), Cate gory Theory (3rd ed.), Heldermann Verlag Berlin,
ISBN 978-3-88538-001-6. 978-3-88538-001-6.
• Kashi Kashiwa wara, ra,
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• Lawvere, F. William; Rosebrugh, Robert (2003).
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• Adámek, Adámek, Jiří; Herrlich, Herrlich, Horst; Strecker, George E.
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Working Wor king Mathematician. Graduate Graduate Texts Texts in Mathematics 5 (2nd ed.). Springer-Verlag. Springer-Verlag. ISBN ISBN 0-38798403-8.. MR 1712872. 98403-8 1712872.
• Mac Mac
Lane, Lane, Saund Saunder ers; s; Birkho Birkhoff, ff, Garre Garretttt (19 (1999) 99) [1967]. Algebra (2nd ed.). Chelsea. ISBN Chelsea. ISBN 0-82181646-2.. 1646-2
• Mart Martin ini,i,
• Bucur, Ion; Deleanu, Aristide (1968). Introduction • Freyd, Peter J. (1964). Abelian Categories.
• Leinster Leinster,,
New New
• Freyd, Peter J.; Scedrov, Andre (1990). Categories,
allegories . North Holland Mathematical Mathematical Library 39. North Holland. ISBN Holland. ISBN 978-0-08-088701-2. 978-0-08-088701-2 .
A.; A.; Ehri Ehrig, g, H.; H.; Nune Nunes, s, D. (199 (1996) 6).. “Elements of basic category theory”. theory” . Technical Re port (Technical (Technical University Berlin) 96 (5).
• May, Peter (1999). A Concise Course in Algebraic Topology. University University of Chicago Chicago Press. ISBN 0-
226-51183-9.. 226-51183-9
• Guerino, Guerino, Mazzola Mazzola (2002 (2002).). TheTopos TheTopos of Music, Music, GeoGeo-
metric Logic of Concepts, Theory, and Performance .
Birkhäuser. ISBN Birkhäuser. ISBN 3-7643-5731-2. 3-7643-5731-2.
13
2.13. EXTERNAL EXTERNAL LINKS
• Pedic Pedicchi chio, o,
Maria Maria Cristina Cristina;; Tholen, Tholen, Wa Walter lter,, eds.
(2004). Categorical foundations. foundations. Special topics topics in order, topology, algebra, and sheaf theory . EncyEncyclopedia of Mathematics and Its Applications 97. Cambridge: Cambridge University Press. Press . ISBN 0521-83414-7.. Zbl 1034.18001. 521-83414-7 1034.18001 .
• Pierce, Benjamin C. (1991). Basic Category Theory forr Computer Scientists . MIT Press. fo Press. ISBN 978-0-
262-66071-6.. 262-66071-6
• Schalk, A.; Simmons, H. (2005). An introduction
to Category Theory in four easy movements (PDF). Notes for for a course offere offeredd as part of the MSc. in Mathematical Logic, Logic, Manchester University. University.
• Simpson, Carlos. Homotopy theory of higher cate gories. arXiv arXiv::1001.4071 1001.4071.,., draft of a book.
• Taylor, Paul (1999). Pr Practic actical al Fo Foundat undations ions of Mat Mathh-
ematics. Cambrid Cambridge ge Studies Studies in Advanced Advanced MatheMathe-
matics 59. Cambridge University Press. Press. ISBN ISBN 9780-521-63107-5.. 0-521-63107-5
• Turi, Daniele (1996–2001). “ (1996–2001). “Cate Category gory Theo Theory ry Lecture Notes” (PDF). Notes” (PDF). Retrieved 11 December 2009. Based on Mac on Mac Lane 1998. 1998 .
2.12 2.12
Furth Further er read readin ing g
• Jean-Pierre Marquis (2008). From a Geometrical Point Point of View: View: A Stud Study of the Histor Historyy and Philo Philosop sophy hy of Category Theory . Springer Science Science & Business Business
Media. ISBN Media. ISBN 978-1-4020-9384-5. 978-1-4020-9384-5 .
2.13 2.13
Exte Ex tern rnal al link linkss
• Theory and Application of Categories, Categories , an electronic journal of category theory, full text, free, since 1995.
• nLab nLab,, a wiki project on mathematics, physics and
philosophy with emphasis on the n-categorical point of view.
• André AndréJoy Joyal al,, CatLab CatLab,, a wiki wiki pro project ject dedi dedica cated ted to the exposition of categorical mathematics.
• Category Theory, Theory, a web page of links to lecture notes and freely available books on category theory.
• Hillman, Chris, A Categorical Primer , CiteSeerX CiteSeerX:: 10.1.1 10.1 .1.24 .24.3264 .3264,, a formal introduction to category theory.
• Adamek, J.; Herrlich, H.; Stecker, G. “Abstract G. “Abstract and Concrete Categories-The Joy of Cats” (PDF). Cats” (PDF).
• Marquis, Jean-Pierre. “Category Theory”. Theory”. Stanford Encyclopedia of Philosophy . with an extensive bib-
liography.
• List of academic conferences on category theory • Baez, John (1996). “The Tale of n-categories” -categories”.. — An informal introduction to higher order categories.
• WildCats is
a categ ategor oryy theo theory ry pac package age for Mathematica.. Manipula Mathematica Manipulation tion and and visualiz visualizatio ationn of objects, objects, morphisms, morphisms, categories, functors, functors, natural transformations,, universal properties. transformations properties.
• The catsters’s channel on channel on YouTube YouTube,, a channel about category theory.
• Category Theory Theory at at PlanetMath.or PlanetMath.orgg. • Video archive of archive of recorded talks relevant to categories, logic and the foundations of physics.
• Interactive Web page which page which generates examples of
categorical constructions in the category of finite sets.
• Category Theory for the Sciences, Sciences , an instruction on category theory as a tool throughout the sciences.
Chapter 3
Field This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation).
integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, algebraic function fields, algebraic number fields, p-adic fields, and so forth. Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations. In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry. As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2 x = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative. A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field . (Historically, division rings were sometimes referred to as fields, while fields were called commutative fields .) As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions: commutative rings
3.1
Definition and illustration
Intuitively, a field is a set F that is a commutative group with respect to two compatible operations, addition and multiplication (the latter excluding zero), with “compatible” being formalized by distributivity, and the caveat that the additive and the multiplicative identities are distinct (0 ≠ 1). The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication , and denoted by + and ·, respectively, such that the following axioms hold (note that subtraction and division aredefined in terms of theinverse operations of addition and multiplication): [note 1] For all a, b in F , both a + b and a · b are in F (or more formally, + and · are binary operations on F ).
Closure of F under addition and multiplication
Associativity of addition and multiplication For
all a , b, and c in F , the following equalities hold: a + (b + c ) = (a + b) + c and a · (b · c ) = (a · b) · c .
Commutativity of addition and multiplication For
all a and b in F , the following equalities hold: a + b = b + a and a · b = b · a.
Existence of additive and multiplicative identity elements
There exists an element of F , called the additive identity element and denoted by 0, such that for all a in F , a + 0 = a . Likewise, there is an element, called the multiplicative identity element and denoted by 1, such that for all a in F , a · 1 = a . To exclude the trivial ring, the additive identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses
⊃ integral domains ⊃ 14
For every a in F , there exists an element − a in F , such that a + (−a) = 0. Similarly, for any a in F other than 0, there exists an element a−1 in F , such
15
3.2. RELATED ALGEBRAIC STRUCTURES
that a · a −1 = 1. (The elements a + (−b) and a · b−1 are also denoted a − b and a/b, respectively.) In other words, subtraction and division operations exist.
The following example is a field consisting of four elements called O, I, A and B. The notation is chosen such that O plays the role of the additive identity element (denoted 0 in the axioms), and I is the multiplicative identity (denoted 1 above). One can check that all field axioms Distributivity of multiplication over addition For all are satisfied. For example: a, b and c in F , the following equality holds: a · ( b + c ) = (a · b) + (a · c ). A·(B+A)=A·I=A,whichequalsA·B+A· A = I + B = A, as required by the distributivity. A field is therefore an algebraic structure F , +, ·, −, −1 , 0, 1; of type 2, 2, 1, 1, 0, 0, consisting of two abelian The above field is called a finite field with four elements, groups: and can be denoted F 4 . Field theory is concerned with understanding the reasons for the existence of this field, • F under +, −, and 0; defined in a fairly ad-hocmanner, and describing its inner structure. For example, from a glance at the multiplica• F ∖ {0} under ·, −1 , and 1, with 0 ≠ 1, tion table, it can be seen that any non-zero element (i.e., I, A, and B) is a power of A: A = A 1, B = A2 = A · A, and [1] with · distributing over +. finally I = A 3 = A · A · A. This is not a coincidence, but rather one of thestarting pointsof a deeper understanding of (finite) fields. 3.1.1 First example: rational numbers A simple example of a field is the field of rational numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers, and b ≠ 0. The additive inverse of such a fraction is simply − a/b, and the multiplicative inverse (provided that a ≠ 0) is b/a. To see the latter, note that
3.1.3
Alternative axiomatizations
As with other algebraic structures, there exist alternative axiomatizations. Because of the relations between the operations, one can alternatively axiomatize a field by explicitly assuming that there are four binary operations (add, subtract, multiply, divide) with axioms relating these, or (by functional decomposition) in terms of b a ba two binary operations (add and multiply) and two unary · = ab = 1 . a b operations (additive inverse and multiplicative inverse), The abstractly required field axioms reduce to stan- or other variants. dard properties of rational numbers, such as the law of The usual axiomatization in terms of the two operations distributivity of addition andmultiplication is briefandallows theother operations to be defined in terms of these basic ones, but in other contexts, such as topology and category thea c e ory, it is important to include all operations as explicitly + · given, rather than implicitly defined (compare topological b d f group). This is because without further assumptions, a c f e d the implicitly defined inverses may not be continuous (in = · + · · topology), or may not be able to be defined (in category b d f f d theory). Defining an inverse requires that one is working a cf ed a cf + ed with a set, not a more general object. = · + = · b df fd b df For a very economical axiomatization of the field of real a(cf + ed) acf aed ac ae numbers, whose primitives are merely a set R with 1 ∈ R, = = + = + bdf bdf bdf bd bf addition, and a binary relation, "<". See Tarski’s axiomatization of the reals. a c a e = · + · ,
� � � � � �
b
d
b
f
or the law of commutativity and law of associativity. 3.1.2
3.2
Related algebraic structures
Second example: a field with four el- The axioms imposed above resemble the ones familiar from other algebraic structures. For example, the exisements tence of the binary operation "·", together with its comIn addition to familiar number systems such as the ratio- mutativity, associativity, (multiplicative) identity element nals, there are other, less immediate examples of fields. andinverses arepreciselytheaxioms for an abelian group.
16
CHAPTER 3. FIELD
In other words, for any field, the subset of nonzero elements F \ {0}, also often denoted F × , is an abelian group (F × , ·) usually called multiplicative group of the field. Likewise (F , +) is an abelian group. The structure of a field is hence thesame as specifying suchtwogroup structures (on the same set), obeying the distributivity. Important other algebraic structures such as rings arise when requiring only part of the above axioms. For example, if the requirement of commutativity of the multiplication operation · is dropped, one gets structures usually called division rings or skew fields.
Ernst Steinitz published the very influential paper Algebraische Theorie der Körper (English: Algebraic Theory of Fields).[8] In this paper he axiomatically studies the properties of fields and defines many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension. Emil Artin developed the relationship between groups and fields in great detail from 1928 through 1942.
3.2.1
3.4.1
Remarks
By elementary group theory, applied to theabelian groups (F × , ·), and (F , +), the additive inverse − a and the multiplicative inverse a−1 are uniquely determined by a. Similar direct consequences from the field axioms include −(a · b) = (−a) · b = a · (−b), in particular −a = (−1) · a as well as a · 0 = 0.
Both can be shown by replacing b or c with 0 in the distributive property.
3.4
Examples Rationals and algebraic numbers
The field of rational numbers Q has been introduced above. A related class of fields very important in number theory are algebraic number fields. We will first give an example, namely the field Q(ζ) consisting of numbers of the form a + bζ
with a , b ∈ Q , where ζ is a primitive third root of unity, i.e., a complex number satisfying ζ 3 = 1, ζ ≠ 1. This field extension can be used to prove a special case of Fermat’s last theorem, which asserts the non-existence of rational nonzero solutions to the equation x 3 + y3 = z3 .
In the language of field extensions detailed below, Q (ζ) is a field extension of degree 2. Algebraic number fields are by definition finite field extensions of Q, that is, fields The concept of field was used implicitly by Niels Henrik containing Q having finite dimension as a Q-vector space. Abel and Évariste Galois in their work on the solvability of polynomial equations with rational coefficients of 3.4.2 Reals, complex numbers, and p-adic degree five or higher. numbers In1857, Karl von Staudtpublished his Algebra of Throws which provided a geometric model satisfying the axioms of a field.[2] This construction has been frequently re- Take the real numbers R , under the usual operations of called as a contribution to the foundations of mathemat- addition and multiplication. When the real numbers are given the usual ordering, they form a complete ordered ics. field ; it is this structure which provides the foundation for In 1871, Richard Dedekind introduced, for a set of real most formal treatments of calculus. or complex numbers which is closed under the four arithmetic operations, the German word Körper , which means The complex numbers C consist of expressions “body” or “corpus” (to suggest an organically closed a + bi entity),[3] hence the common use of the letter K to denote a field. He also defined rings (then called order or ordermodul ), but the term “a ring” ( Zahlring) was invented by where i is the imaginary unit, i.e., a (non-real) number Hilbert.[4] In 1893, Eliakim Hastings Moore called the satisfying i2 = −1. Addition and multiplication of real concept “field” in English.[5][6] numbers are defined in such a way that all field axioms In1881, Leopold Kronecker definedwhathecalleda“do- hold for C. For example, the distributive law enforces main of rationality”, which is indeed a field of polynomials in modern terms. In 1893, Heinrich M. Weber gave (a + bi)·(c + d i) = ac + bc i + ad i + bd i2 , which [7] the first clear definition of an abstract field. In 1910, equals ac −bd + (bc + ad )i.
3.3
History
17
3.4. EXAMPLES
The real numbers can be constructed by completing the rationalnumbers, i.e., filling the“gaps": for example √2 is such a gap. By a formally verysimilar procedure, another important class of fields, the field of p-adic numbers Q p is built. It is used in number theory and p-adic analysis. Hyperrealnumbers and superreal numbers extend the real numbers with the addition of infinitesimal and infinite numbers. 3.4.3
Constructible numbers
ple F 4 is a field with four elements. F2 consists of two elements, 0 and 1. This is the smallest field, because by definition a field has at least two distinct elements 1 ≠ 0. Interpreting the addition and multiplication in this latter field as XOR and AND operations, this field finds applications in computer science, especially in cryptography and coding theory. In a finite field there is necessarily an integer n such that 1 + 1 + ··· + 1 ( n repeated terms) equals 0. It can be shown that the smallest such n must be a prime number, called the characteristic of the field. If a (necessarily infinite) field has the property that 1 + 1 + ··· + 1 is never zero, for any number of summands, such as in Q, for example, the characteristic is said to be zero. A basic class of finite fields are the fields F p with p elements ( p a prime number): F p = Z/ pZ = {0, 1, ..., p − 1},
Given 0, 1, r1 and r2 , the construction yields r1 ·r2
In antiquity, several geometric problems concerned the (in)feasibility of constructing certain numbers with compass and straightedge. For example, it was unknown to the Greeks that it is in general impossible to trisect a given angle. Using the field notion and field theory allows these problems to be settled. To do so, the field of constructible numbers is considered. It contains, on the plane, the points 0 and 1, and all complex numbers that can be constructed from these two by a finite number of construction steps using only compass and straightedge. This set, endowed with the usual addition and multiplication of complex numbers does form a field. For example, multiplying two (real) numbers r 1 and r 2 that have already been constructed can be done using construction at the right, based on the intercept theorem. This way, the obtained field F contains all rational numbers, but is bigger than Q, because for any f ∈ F , the square root of f is also a constructible number. A closely related concept is that of a Euclidean field, namely an ordered field whose positive elements are closed under square root. The real constructible numbers formtheleast Euclidean field, andtheEuclidean fields are precisely the ordered extensions thereof.
where the operations are defined by performing the operation in the set of integers Z, dividing by p and taking the remainder; see modular arithmetic. A field K of characteristic p necessarily contains F p,[9] and therefore may be viewed as a vector space over F p, of finite dimension if K is finite. Thus a finite field K has prime power order, i.e., K has q = pn elements (where n > 0 is the number of elements in a basis of K over F p). By developing more field theory, in particular the notion of the splitting field of a polynomial f over a field K , which is thesmallest field containing K and all roots of f , one can show that two finite fields with the same number of elements are isomorphic, i.e., there is a one-to-one mapping of one field onto the other that preserves multiplication and addition. Thus we may speak of the finite field with q elements, usually denoted by Fq or GF(q).
3.4.5
Archimedean fields
Main article: Archimedean field
An Archimedean field is an ordered field such that for each element there exists a finite expression 1 + 1 + ··· + 1 whose value is greater than that element, that is, there are no infinite elements. Equivalently, the field contains no infinitesimals; or, the field is isomorphic to a subfield of the reals. A necessary condition for an ordered field to be complete is that it be Archimedean, since in any non3.4.4 Finite fields Archimedean field there is neither a greatest infinitesimal nora leastpositiverational, whencethesequence1/2, 1/3, Main article: Finite field 1/4, …, every element of which is greater than every infinitesimal, has no limit. (And since every proper subfield Finite fields (also called Galois fields) are fields with of the reals also contains such gaps, up to isomorphism finitely many elements. The above introductory exam- the reals form the unique complete ordered field.)
18 3.4.6
CHAPTER 3. FIELD
Field of functions
Given a geometric object X , one can consider functions on such objects. Adding and multiplying them pointwise, i.e., ( f ⋅ g )( x ) = f ( x ) ⋅ g ( x ) this leads to a field. However, for having multiplicative inverses, one has to consider partial functions, which, almost everywhere, are defined and have a non-zero value. If X is an algebraic variety over a field F , then the rational functions X → F form a field, the function field of X . This field consists of the functions that are defined and are the quotient of two polynomial functions outside some subvariety. Likewise, if S is a Riemann surface, then the meromorphic functions S → C form a field. Under certain circumstances, namely when S is compact, S can be reconstructed from this field. 3.4.7
Local and global fields
Another important distinction in the realm of fields, especially with regard to number theory, are local fields and global fields. Local fields are completions of global fields at a given place. For example, Q is a global field, and the attached local fields are Q p and R (Ostrowski’s theorem). Algebraic number fields and function fields over Fq are further global fields. Studyingarithmetic questions in global fields may sometimes be done by looking at the corresponding questions locally—this technique is called local-global principle.
and is algebraically closed, i.e., any such polynomial does have at least one solution in F . The algebraic closure is unique up to isomorphism inducing the identity on F . However, in many circumstances in mathematics, it is not appropriate to treat F as being uniquely determined by F , since the isomorphism above is not itself unique. In these cases, one refers to such a F as an algebraic closure of F . A similar concept is the separable closure, containing all roots of separable polynomials, instead of all polynomials. For example, if F = Q , the algebraic closure Q is also called field of algebraic numbers . The field of algebraic numbers is an example of an algebraically closed field of characteristic zero; as such it satisfies the same first-order sentences as the field of complex numbers C. In general, all algebraic closures of a field are isomorphic. However, there is in general no preferable isomorphism between two closures. Likewise for separable closures. 3.6.2
Subfields and field extensions
A subfield is, informally, a small field contained in a bigger one. Formally, a subfield E of a field F is a subset containing 0 and1, closed under theoperations +, −, · and multiplicative inverses and with its own operations defined by restriction. For example, the real numbers contain several interesting subfields: the real algebraic numbers, the computable numbers and the rational numbers are examples. The notion of field extension lies at the heart of field theory, and is crucial to many other algebraic domains. A 3.5 Some first theorems field extension F / E is simply a field F and a subfield E ⊂ F . Constructing such a field extension F / E can be done F × • Every finite subgroup of the multiplicative group by “adding new elements” or adjoining elements to the is cyclic. This applies in particular to Fq× , it is field E . For example, given a field E , the set F = E (X ) of cyclic of order q − 1. In the introductory example, rational functions, i.e., equivalence classes of expressions a generator of F4 × is the element A. of the kind • An integral domain is a field if and only if it has no ideals except{0} and itself. Equivalently, an integral domain is a field if and only if its Krull dimension is p(X ) , q (X ) 0. where p(X ) and q(X ) are polynomials with coefficients in E , and q is not the zero polynomial, forms a field. This is the simplest example of a transcendental extension of E . It also is an example of a domain (the ring of polynomials 3.6 Constructing fields E in this case) being embedded into its field of fractions E (X ) . 3.6.1 Closure operations The ring of formal power series E [[X ]] is also a domain, Assuming the axiom of choice, for every field F , there and again the (equivalence classes of) fractions of the exists a field F , called the algebraic closure of F , which form p(X )/ q(X ) where p and q are elements of E [[X ]] contains F , is algebraic over F , which means that any el- form the field of fractions for E [[X ]] . This field is actually the ring of Laurent series over the field E , denoted ement x of F satisfies a polynomial equation E ((X )) . fnx n + fn₋₁ x n−1 + ··· + f 1 x + f 0 = 0, with coIn theabovetwocases, theadded symbol X anditspowers efficients fn, ..., f 0 ∈ F , did not interact with elements of E . It is possible however
• Isomorphism extension theorem
19
3.7. GALOIS THEORY
that the adjoined symbol may interact with E . This idea will be illustrated by adjoining an element to the field of real numbers R. As explained above, C is an extension of R. C can be obtained from R by adjoining the imaginary symbol i which satisfies i 2 = −1. The result is that R[i]=C. This is different from adjoining the symbol X to R , because in that case, the powers of X are all distinct objects, but here, i2 =−1 is actually an element of R. Another way to view this last example is to note that i is a zero of the polynomial p (X ) = X 2 + 1. The quotient ring R[X ]/(X 2 + 1) can be mapped onto C using the map a + bX → a + ib . Since the ideal (X 2 +1) is generated by a polynomial irreducible over R, the ideal is maximal, hence the quotient ring is a field. This nonzero ring map from the quotient to C is necessarily an isomorphism of rings. The above construction generalises to any irreducible polynomial in the polynomial ring E [X ], i.e., a polynomial p(X ) that cannot be written as a product of nonconstant polynomials. The quotient ring F = E [X ] / ( p(X )), is again a field. Alternatively, constructing such field extensions can also be done, if a bigger container is already given. Suppose given a field E , andafield G containing E as a subfield, for example G could be thealgebraic closureof E . Let x bean element of G not in E . Then there is a smallest subfield of G containing E and x , denoted F = E ( x ) and called field extension F / E generatedby x in G .[10] Suchextensions are also called simple extensions . Many extensions are of this type; see the primitive element theorem. For instance, Q(i ) is the subfield of C consisting of all numbers of the form a + bi where both a and b are rational numbers. One distinguishes between extensions having various qualities. Forexample, an extension K ofafield k is called algebraic , if every element of K is a root of some polynomial with coefficients in k . Otherwise, the extension is called transcendental . The aim of Galois theory is the study of algebraic extensions of a field.
3.6.3
Rings vs fields
Adding multiplicative inverses to an integral domain R yields the field of fractions of R. For example, the field of fractions of the integers Z is just Q. Also, the field F (X ) is the quotient field of the ring of polynomials F [X ]. Anothermethodtoobtainafieldfromacommutativering R is taking the quotient R / m , where m is any maximal ideal of R. Theabove construction of F = E [X ] / ( p(X ) ),is an example, because the irreducibility of the polynomial p(X ) is equivalent to themaximality of theideal generated by this polynomial. Another example are the finite fields F p = Z / pZ.
3.6.4
Ultraproducts
If I is an index set, U is an ultrafilter on I , and Fi is a field for every i in I , the ultraproduct of the Fi with respect to U is a field. For example, a non-principal ultraproduct of finite fields is a pseudo finite field; i.e., a PAC field having exactly one extension of any degree.
3.7 Galois theory Main article: Galois theory Galois theory aims to study the algebraic extensions of a field by studying the symmetry in the arithmetic operations of addition and multiplication. The fundamental theorem of Galois theory shows that there is a strong relation between the structure of the symmetry group and the set of algebraic extensions. In the case where F / E is a finite (Galois) extension, Galois theory studies the algebraic extensions of E that are subfields of F . Such fields are called intermediate extensions. Specifically, the Galois group of F over E , denoted Gal(F /E ), is the group of field automorphisms of F that aretrivial on E (i.e., the bijections σ : F → F that preserve addition and multiplication and that send elements of E to themselves), and the fundamental theorem of Galois theory states that there is a one-to-one correspondence between subgroups of Gal(F /E ) and the set of intermediate extensions of the extension F /E . The theorem, in fact, gives an explicit correspondence and further properties. To study all (separable) algebraic extensions of E at once, one must consider the absolute Galois group of E , defined as the Galois group of the separable closure, E sep , of E over E i.e., Gal(E sep /E ). It is possible that the degree of this extension is infinite (as in the case of E = Q ). It is thus necessary to have a notion of Galois group for an infinite algebraic extension. The Galois group in this case is obtained as a “limit” (specifically an inverse limit) of the Galois groups of the finite Galois extensions of E . In this way, it acquires a topology.[note 2] The fundamental theorem of Galois theory can be generalized to the case of infinite Galois extensions by taking into consideration the topology of the Galois group, and in the case of E sep/E it states that there this a one-to-one correspondence between closed subgroups of Gal(E sep /E ) and the set of all separable algebraic extensions of E (technically, one only obtains those separable algebraic extensions of E that occur as subfields of the chosen separable closure E sep , but since all separable closures of E are isomorphic, choosing a different separable closure would give the same Galois group and thus an “equivalent” set of algebraic extensions).
20
CHAPTER 3. FIELD
3.8
Generalizations
Finite fields are used in number theory, Galois theory, cryptography, coding theory and combinatorics; and There are also proper classes with field structure, which again the notion of algebraic extension is an important tool. are sometimes called Fields, with a capital F:
• The surreal numbers form a Field containing the re-
als, and would be a field except for the fact that they 3.10 See also are a proper class, not a set. • Category of fields The nimbers form a Field. The set of nimbers with n • Glossary of field theory for more definitions in field birthday smaller than 2 2 , the nimbers with birthday theory. smaller than any infinite cardinal are all examples of fields. • Heyting field
•
In a different direction, differential fields are fields • Lefschetz principle equipped with a derivation. For example, the field • Puiseux series R(X ), together with the standard derivative of polynomials forms a differential field. These fields are central to • Ring differential Galois theory. Exponential fields, meanwhile, are fields equipped with an exponential function that pro• Vector space vides a homomorphism between the additive and multiplicative groups within the field. The usual exponential • Vector spaces without fields function makes the real and complex numbers exponential fields, denoted Rₑₓ and Cₑₓ respectively. Generalizing in a more categorical direction yields the 3.11 Notes field with one element and related objects. 3.8.1
Exponentiation
[1] That is, the axiom for addition only assumes a binary operation + : F × F → F , a , b → a + b. The axiom of inverse allows one to define a unary operation − : F → F a → −a that sends an element to its negative (its additive inverse); this is not taken as given, but is implicitly defined in terms of addition as " −a is the unique b such that a + b = 0 ", “implicitly” because it is defined in terms of solving an equation—and one then defines the binary operation of subtraction, also denoted by "−", as − : F × F → F, a, b → a − b := a + (−b) in terms of addition and additive inverse. In the same way, one defines the binary operation of division ÷ in terms of the assumed binary operation of multiplication and the implicitly defined operation of “reciprocal” (multiplicative inverse).
One does not in general study generalizations of fields with three binary operations. The familiar addition/subtraction, multiplication/division, exponentiation/root-extraction/logarithm operations from the natural numbers to the reals, each built up in terms of iteration of the last, mean that generalizing exponentiation as a binary operation is tempting, but has generally not proven fruitful; instead, an exponential field assumes a unary exponential function from the additive group to the multiplicative group, not a partially defined binary function. Note that the exponential operation [2] As an inverse limit of finite discrete groups, it is equipped of a is neither associative nor commutative, nor has a with the profinite topology, making it a profinite topologunique inverse ( ± 2 are both square roots of 4, for inical group stance), unlike addition and multiplication, and further is not defined for many pairs—for example, ( −1)1/2 = √ −1 does not define a single number. These all show that 3.12 References even for rational numbers exponentiation is not nearly as well-behaved as addition and multiplication, which is [1] Wallace, D A R (1998) Groups, Rings, and Fields, SUMS. why one does not in general axiomatize exponentiation. b
Springer-Verlag: 151, Th. 2.
3.9 Applications The concept of a field is of use, for example, in defining vectors and matrices, two structures in linear algebra whose components can be elements of an arbitrary field.
[2] Karl Georg Christian v. Staudt, Beiträge zur Geometrie der Lage (Contributions to the Geometry of Position), volume 2 (Nürnberg, (Germany): Bauer and Raspe, 1857). See: “Summen von Würfen” (sums of throws), pp. 166171 ; “Produckte aus Würfen” (products of throws), pp. 171-176 ; “Potenzen von Würfen” (powers of throws), pp. 176-182.
21
3.14. EXTERNAL LINKS [3] Peter Gustav Lejeune Dirichlet with R. Dedekind, Vorlesungen über Zahlentheorie von P. G. Lejeune Dirichlet
3.14
External links
(Lectures on Number Theory by P.G. Lejeune Dirichlet), 2nd ed., volume 1 (Braunschweig, Germany: Friedrich Vieweg und Sohn, 1871), p. 424. From page 424: “Unter einem Körper wollen wir jedes System von unendlich vie-
• Hazewinkel,
len reellen oder complexen Zahlen verstehen, welches in sich so abgeschlossen und vollständig ist, dass die Addition, Subtraction, Multiplication und Division von je zwei dieser Zahlen immer wieder eine Zahl desselben Systems hervorbringt.” (By a “field” we will understand any sys-
• Field Theory Q&A • Fieldsat ProvenMath definition andbasic properties. • Field at PlanetMath.org.
tem of infinitely many real or complex numbers, which is so closed and complete that the addition, subtraction, multiplication, and division of any two of these numbers always again produces a number of the same system.) [4] J J O'Connor and E F Robertson, The development of Ring Theory, September 2004. [5] Moore, E. Hastings (1893), “A doubly-infinite system of simple groups”, Bulletin of the New York Mathematical Society 3 (3): 73–78, doi:10.1090/S0002-9904-189300178-X, JFM 25.0198.01. From page 75: “Such a system of s marks [i.e., a finite field with s elements] we call a field of order s.” [6] Earliest Known Uses of Some of the Words of Mathematics (F)
[7] Fricke, Robert; Weber, Heinrich Martin (1924), Lehrbuch der Algebra, Vieweg, JFM 50.0042.03 [8] Steinitz, Ernst (1910), “Algebraische Theorie der Körper”, Journal für die reine und angewandte Mathematik 137: 167–309, doi:10.1515/crll.1910.137.167, ISSN 0075-4102, JFM 41.0445.03 [9] Jacobson (2009), p. 213 [10] Jacobson (2009), p. 213
3.13
Sources
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2, especially Chapter 13
• Allenby, R.B.J.T. (1991), Rings, Fields and Groups , Butterworth-Heinemann, ISBN 978-0-340-544402
• Blyth, T.S.; Robertson, E. F. (1985), Groups, rings
and fields: Algebra through practice, Cambridge
University Press. See especially Book 3 (ISBN 0521-27288-2) and Book 6 (ISBN 0-521-27291-2).
• Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1
• James Ax (1968), The elementary theory of finite fields, Ann. of Math. (2), 88, 239–271
Michiel, ed.
(2001), “Field”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Chapter 4
Galois group In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with thefield extension. Thestudyof field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them. For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.
• Aut(R/Q) is trivial.
Indeed, it can be shown that any automorphism of R must preserve the ordering of the real numbers and hence must be the identity.
• Aut(C/Q) is an infinite group. • Gal(Q(√2)/Q) has two elements, the identity auto-
morphism and the automorphism which exchanges √2 and −√2.
• Consider the field K = Q(³√2). The group Aut(K/Q) contains only the identity automorphism. This is because K is not a normal extension, since the other two cube roots of 2 (both complex) aremissing from the extension — in other words K is not a splitting field.
4.1 Definition Suppose that E is an extension of the field F (written as E /F and read E over F ). An automorphism of E /F is defined to be an automorphism of E that fixes F pointwise. In other words, an automorphism of E /F is an isomorphism α from E to E such that α( x ) = x for each x in F . The set of all automorphisms of E /F forms a group with the operation of function composition. This group is sometimes denoted by Aut(E /F ). If E /F is a Galois extension, then Aut(E /F ) is called the Galois group of (the extension) E over F , and is usually denoted by Gal(E /F ).[1] If E /F is not a Galois extension, then the Galois group of (the extension) E over F is sometimes defined as Aut(G /F ), where G is the Galois closure of E .
4.2
Examples
• Consider now L = Q(³√2, ω), where ω is a primitive third root of unity. The group Gal(L/Q) is isomorphic to S 3 , the dihedral group of order 6, and L is in fact the splitting field of x 3 − 2 over Q.
• If q isn a prime power, and if F = GF(q) and E =n GF(q
) denote the Galois fields of order q and q respectively, then Gal( E /F ) is cyclic of order n and generated by the Frobenius homomorphism.
• If f is an irreducible polynomial of prime degree p with rational coefficients and exactly two nonreal roots, then the Galois group of f is the full symmetric group Sp .
For a finite field Fq , wealways haveGal(Fq /Fq ) cyclic of order n , generated by the q th power Frobenius automorphism. n
n
In the following examples F is a field, and C , R , Q are 4.3 Properties the fields of complex, real, and rational numbers, respectively. The notation F (a) indicates the field extension obThe significance of an extension being Galois is that it tained by adjoining an element a to the field F . obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of • Gal(F /F ) is the trivial group that has a single ele- the Galois group correspond to the intermediate fields of ment, namely the identity automorphism. the field extension. • Gal(C/R) has two elements, the identity au- If E /F is a Galois extension, then Gal( E /F ) can be given tomorphism and the complex conjugation a topology, called the Krull topology, that makes it into a automorphism. [2] profinite group. 22
4.7. EXTERNAL LINKS
4.4
See also
• Absolute Galois group 4.5 Notes [1] Some authors refer to Aut(E /F ) as the Galois group for arbitrary extensions E /F and use the corresponding notation, e.g. Jacobson 2009. [2] Cooke, Roger L. (2008), Classical Algebra: Its Nature, Origins, and Uses, John Wiley & Sons, p. 138, ISBN 9780470277973.
4.6 References
• Jacobson, Nathan (2009) [1985], Basic algebra I (Second ed.), Dover Publications, ISBN 978-0-48647189-1
• Lang, Serge (2002), Algebra, Graduate Texts in
Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
4.7 External links
• Hazewinkel, Michiel, ed.
(2001), “Galois group”, Encyclopedia of Mathematics , Springer, ISBN 9781-55608-010-4
• “Galois Groups” at MathPages.com.
23
Chapter 5
Group This article is about basic notions of groups in mathemat- changed and the operation of combining two such transics. For a more advanced treatment, see Group theory. formations by performing one after the other. Lie groups In mathematics, a group is an algebraic structure con- are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and Point groups are used to help understand symmetry phenomena in molecular chemistry . The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline— studies groups in their own right. a[›] To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely (its group representations), both from a theoretical and a computational point of view. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed The manipulations of this Rubik’s Cube form the Rubik’s Cube in 2004.aa[›] Since the mid-1980s, geometric group the group. ory, which studies finitely generated groups as geometric sisting of a set of elements equipped with an operation objects, has become a particularly active area in group that combines any two elements to form a third ele- theory. ment. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a 5.1 Definition and illustration group is the set of integers together with the addition operation, but the abstract formalization of the group ax- 5.1.1 First example: the integers ioms, detached as it is from the concrete nature of any particular group and its operation, applies much more One of the most familiar groups is the set of integers Z widely. It allows entities with highly diverse mathemati- which consists of the numbers cal origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ..., [3] together aspects. The ubiquity of groups in numerous areas within with addition. and outside mathematics makes them a central organizing principle of contemporary mathematics. [1][2] The following properties of integer addition serve as a Groups share a fundamental kinship with the notion of modelfor theabstractgroup axioms given in thedefinition symmetry. For example, a symmetry group encodes sym- below. metry features of a geometrical object: the group con• For any two integers a and b, the sum a + b is also sists of the set of transformations that leave the object un24
5.1. DEFINITION AND ILLUSTRATION
25
a • b = b • a an integer. That is, addition of integersalways yields an integer. This property is known as closure under addition. may not always be true. This equation always holds in the • For all integers a, b and c , (a + b ) + c = a + (b + group of integers under addition, because a + b = b + a c ). Expressed in words, adding a to b first, and then for any two integers (commutativity of addition). Groups adding the result to c gives the same final result as for which the commutativity equation a • b = b • a always adding a to the sum of b and c , a property known as holds are called abelian groups (in honor of Niels Henrik Abel). The symmetry group described in the following associativity. is an example of a group that is not abelian. • If a is any integer, then 0 + a = a + 0 = a. Zero section is called the identity element of addition because The[6]identity element of a group G is often written as 1 or 1G , a notation inherited from the multiplicative idenadding it to any integer returns the same integer. tity. If a group is abelian, then one may choose to denote • For every integer a, there is an integer b such that a the group operation by + and the identity element by 0; + b = b + a = 0. The integer b is called the inverse in that case, the group is called an additive group. The element of the integer a and is denoted − a. identity element can also be written as id . The integers, together with the operation +, form a math- The set G is called the underlying set of the group (G , •). ematical object belonging to a broad class sharing simi- Often thegroup’s underlying set G is used as a short name lar structural aspects. To appropriately understand these for the group (G , •). Along the same lines, shorthand exstructures as a collective, the following abstract definition pressions suchas “a subset of the group G " or “an element of group G " are used when what is actually meant is “a is developed. subset of the underlying set G of the group (G , •)" or “an element of theunderlyingset G of the group (G , •)". Usually, it is clear from the context whether a symbol like G 5.1.2 Definition refers to a group or to an underlying set. [T]he axioms for a group are short and natural... Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, 5.1.3 Second example: a symmetry group which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, anything like this exists. Richard Borcherds in Mathematicians: An Outer View of reflections, and translations. Any figureis congruent to itself. However, some figures are congruent to themselves the Inner World [4] in more than one way, and these extra congruences are A group is a set, G , together with an operation • (called called symmetries. A square has eight symmetries. These the group law of G ) that combines any two elements a and are: b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, ( G , •), must satisfy four • the identity operationleaving everything unchanged, requirements known as the group axioms :[5] denoted id; Closure For all a, b in G , the result of the operation, a • • rotations of the square around its center by 90° b, is also in G .b[›] clockwise, 180° clockwise, and 270° clockwise, denoted by r1 , r2 and r3 , respectively; Associativity For all a, b and c in G , (a • b) • c = a • (b • c ). • reflections about the vertical and horizontal middle Identity element There exists an element e in G , such line (f and fᵥ), or through the two diagonals (f and that for every element a in G , the equation e • a = a f). • e = a holds. Such an element is unique (see below), and thus one speaks of the identity element. These symmetries are represented by functions. Each of Inverse element For each a in G , there exists an element these functions sends a point in the square to the corb in G , commonly denoted a −1 (or −a, if the oper- responding point under the symmetry. For example, r 1 ation is denoted "+"), such that a • b = b • a = e , sends a point to its rotation 90° clockwise around the square’s center, andf sends a point to its reflection across where e is the identity element. the square’s vertical middle line. Composing two of these The result of an operation may depend on the order of symmetry functions gives another symmetry function. the operands. In other words, the result of combining These symmetries determine a group called the dihedral element a with element b need not yield the same result group of degree 4 and denoted D 4 . The underlying set as combining element b with element a; the equation of the group is the above set of symmetry functions, and
26
CHAPTER 5. GROUP
the group operation is function composition.[7] Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first a and then b is written symbolically from right to left as b • a (“apply the symmetry b after performing the symmetry a").
The right-to-left notation is the same notation that is used for composition of functions. The group table on the right lists the results of all such compositions possible. For example, rotating by 270° clockwise (r3 ) and then reflecting horizontally (f) is the same as performing a reflection along the diagonal (f). Using the above symbols, highlighted in blue in the group table:
3. The identity element is the symmetry id leaving everything unchanged: for any symmetry a, performing id after a (or a after id) equals a , in symbolic form, id • a = a, a • id = a. 4. An inverse element undoes the transformation of some other element. Every symmetry can be undone: each of the following transformations— identity id, the reflections f, fᵥ, f, f and the 180° rotation r2 —is its own inverse, because performing it twice brings the square back to its original orientation. The rotations r3 and r1 are each other’s inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. In symbols, f • f = id, r3 • r1 = r1 • r3 = id.
f • r3 = f.
Given this set of symmetries and the described operation, In contrast to thegroup of integersabove, where theorder of the operation is irrelevant, it does matter in D 4 : f • r1 the group axioms can be understood as follows: = f but r1 • f = f. In other words, D 4 is not abelian, which makes the group structure more difficult than the 1. The closure axiom demands that the composition b • integers introduced first. a of any two symmetries a and b is also a symmetry. Another example for the group operation is r3 • f = f,
5.2
History
i.e. rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal Main article: History of group theory (f). Indeedeveryother combination of two symmetries still gives a symmetry, as can be checked using The modern concept of an abstract group developed out the group table. of several fields of mathematics. [8][9][10] The original motivation for group theory was the quest for solutions of 2. The associativity constraint deals with composing polynomial equations of degree higher than 4. The 19thmore than two symmetries: Starting with three el- century French mathematician Évariste Galois, extendements a, b and c of D4 , there are two possible ways ing prior work of Paolo Ruffini and Joseph-Louis Laof using these three symmetries in this order to de- grange, gave a criterion for the solvability of a particutermine a symmetry of the square. One of these lar polynomial equation in terms of the symmetry group ways is to first compose a and b into a single sym- of its roots (solutions). The elements of such a Galois metry, then to compose that symmetry with c . The group correspond to certain permutations of the roots. At other way is to first compose b and c , then to com- first, Galois’ ideas were rejected by his contemporaries, pose the resulting symmetry with a. The associativ- and published only posthumously. [11][12] More general ity condition permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the the(a • b) • c = a • (b • c ) n ory of groups, as depending on the symbolic equation θ means that these two ways are the same, i.e., a prod- = 1 (1854) gives the first abstract definition of a finite
uct of many group elements can be simplified in any grouping. For example, (f • fᵥ) • r2 = f • (fᵥ • r2 ) can be checked using the group table at the right While associativity is true for the symmetries of the square and addition of numbers, it is not true for all operations. For instance, subtraction of numbers is not associative: (7 − 3) − 2 = 2 is not the same as 7 − (3 − 2) = 6.
group.[13] Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program.[14] After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.[15]
27
5.4. BASIC CONCEPTS
The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker.[16] In 1847, Ernst Kummer made early attempts to prove Fermat’s Last Theorem by developing groups describing factorization into prime numbers.[17] The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870).[18] Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an “abstract group”, in theterminologyof the time.[19] Asofthe 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers.[20] The theory of Lie groups, and moregenerally locally compactgroups was studied by Hermann Weyl, Élie Cartan and many others.[21] Its algebraic counterpart, thetheoryof algebraic groups,wasfirst shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.[22] The Universityof Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, JohnG.Thompsonand Walter Feit, laying thefoundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher andSmith in 2004. This project exceededprevious mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification. [23] These days, group theory is still a highly active mathematical branch, impacting many other fields. a[›]
actually two-sided, so the resulting definition is equivalent to the one given above.[26] 5.3.1
Uniqueness of identity element and inverses
Two important consequences of the group axioms are the uniqueness of the identity element and the uniqueness of inverse elements. There can be only one identity element in a group, and each element in a group has exactly one inverse element. Thus, it is customary to speak of the identity, and the inverse of an element.[27] To prove the uniqueness of an inverse element of a, suppose that a has two inverses, denoted b and c , in a group (G , •). Then
The two extremal terms b and c are equal, since they are connected by a chain of equalities. In other words, there is only one inverse element of a. Similarly, to prove that the identity element of a group is unique, assume G is a group with two identity elements e and f . Then e = e • f = f , hence e and f are equal. 5.3.2
Division
In groups, the invertibility of the group action means that division is possible: given elements a and b of the group G , there is exactly one solution x in G to the equation x • a = b .[27] In fact, right multiplication of the equation by a−1 gives the solution x = x • a • a−1 = b • a−1 . Similarly there is exactly one solution y in G to the equation a • y = b, namely y = a−1 • b. If the • operation is commutative, we get that x = y. If not, x may be different from y. A consequence of this is that multiplying by a group el g is a bijection. Specifically, if g is an element of 5.3 Elementary consequences of ement the group G , there is a bijection from G to itself called left the group axioms translation by g sending h ∈ G to g • h . Similarly, right translation by g is a bijection from G to itself sending h to Basic facts about all groups that can be obtained directly h • g. If G is abelian, left and right translation by a group from the group axioms are commonly subsumed under element are the same. elementary group theory .[24] For example, repeated applications of the associativity axiom show that the unambiguity of 5.4 Basic concepts a • b • c = (a • b) • c = a • (b • c )
generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within sucha series of terms, parentheses areusuallyomitted.[25] The axioms may be weakened to assert only the existence of a left identityand left inverses. Bothcanbeshowntobe
Further information: Glossary of group theory To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed. c[›] There is a conceptual principle
28
CHAPTER 5. GROUP
underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being “structureless”, do not have), constructions related to groups have to be compatible with the group operation. This compatibility manifests itself in the following notions in various ways. For example, groups can be related to each other via functions called group homomorphisms. By the mentioned principle, they are required to respect the group structures in a precise sense. The structure of groups can also be understood by breaking them into pieces called subgroups and quotient groups. The principle of “preserving structures”—a recurring topic in mathematics throughout—is an instance of working in a category, in this case the category of groups.[28]
In the example above, the identity and the rotations constitute a subgroup R = {id, r1 , r2 , r3 }, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (note that rotation in the opposite direction is not defined). The subgroup test is a necessary and sufficient condition for a nonempty subset H of a group G to be a subgroup: it is sufficient to check that g−1 h ∈ H for all elements g, h ∈ H . Knowing the subgroups is important in understanding the group as a whole.d[›] Given any subset S of a group G , the subgroup generated by S consists of products of elements of S and their inverses. It is thesmallest subgroup of G containing S .[31] In the introductory example above, the subgroup generated 5.4.1 Group homomorphisms by r2 and fᵥ consists of these two elements, the identity element id and f = fᵥ • r 2 . Again, this is a subgroup, beMain article: Group homomorphism cause combining any two of these four elements or their inverses (which are, in this particular case, these same g[›] Group homomorphisms are functions that preserve elements) yields an element of this subgroup. group structure. A function a: G → H between two groups (G , •) and (H , ∗) is called a homomorphism if the 5.4.3 Cosets equation a( g • k ) = a( g) ∗ a(k )
holds for all elements g, k in G . In other words, the result is the same when performing the group operation after or before applying the map a . This requirement ensures that a (1G ) = 1 H , and also a( g)−1 = a( g−1 ) for all g in G . Thus a group homomorphism respects all the structure of G provided by the group axioms. [29] Two groups G and H are called isomorphic if there exist group homomorphisms a : G → H and b : H → G , such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H . That is, a(b(h)) = h and b(a( g)) = g for any g in G and h in H . From an abstract point of view, isomorphic groups carry the same information. For example, proving that g • g = 1G for some element g of G is equivalent to proving that a ( g) ∗ a ( g) = 1H , because applying a to the first equality yields the second, and applying b to the second gives back the first. 5.4.2
Subgroups
Main article: Subgroup Informally, a subgroup is a group H contained within a bigger one, G .[30] Concretely, the identity element of G is contained in H , and whenever h1 and h2 are in H , then so are h1 • h2 and h1 −1 , so the elements of H , equipped with the group operation on G restricted to H , indeed form a group.
Main article: Coset In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D 4 above, once a reflection is performed, thesquare never gets back to ther2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are gH = { g • h : h ∈ H } and Hg = {h • g : h ∈ H },
respectively.[32]
The left cosets of any subgroup H form a partition of G ; thatis,the union ofallleftcosetsisequalto G andtwo left cosets are either equal or have an empty intersection.[33] The first case g1 H = g2 H happens precisely when g1 −1 • g2 ∈ H , i.e. if the two elements differ by an element of H . Similar considerations apply to the right cosets of H . The left and right cosets of H may or may not be equal. If they are, i.e. for all g in G , gH = Hg, then H is said to be a normal subgroup . In D 4 , the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equalto R, if g isanelementof R itself, or otherwiseequal to U = fR = {f, fᵥ, f, f} (highlighted in green). The subgroup R is also normal, because f R = U = R f and
5.5. EXAMPLES AND APPLICATIONS
29
similarly for any element other than f. (In fact, in the surjective maps (every element of the target is mapped case of D4, observe that all such cosets are equal, such onto), such as the canonical map G → G / N .y[›] Interthat fR = fᵥR = fR = fR.) preting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In gen5.4.4 Quotient groups eral, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first Main article: Quotient group isomorphism theorem address this phenomenon. In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or fac- 5.5 Examples and applications tor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N , the quotient Main articles: Examples of groups and Applications of group is defined by group theory G / N = { gN , g ∈ G }, "G modulo N ".[34]
This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G : ( gN ) • ( hN ) = ( gh)N for all g and h in G . This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is ( gN )−1 = ( g−1 )N .e[›] The elements of the quotient group D 4 / R are R itself, which represents the identity, and U = fᵥR. The group operation on thequotientis shown at theright. For example, U • U = fᵥ R • fᵥR = (fᵥ • fᵥ)R = R. Both the subgroup R = {id, r1 , r2, r3 }, as well as the corresponding quotient are abelian, whereas D 4 is not abelian. Building bigger groups by smaller ones, such as D 4 from its subgroup R and the quotient D4 / R is abstracted by a notion called semidirect product. Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4 , for example, can be generated by two elements r and f (for example, r = r1 , the right rotation and f = fᵥ the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations r 4 = f 2 = (r • f )2 = 1,[35]
the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups. Sub- and quotient groupsarerelated in the following way: a subset H of G can be seen as an injective map H → G , i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are
A periodic wallpaper pattern gives rise to a wallpaper group.
The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers. Examples and applications of groups abound. A starting point is the group Z of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group.[36] By means of this connection, topological properties such as proximity and continuity translate into properties of groups. i[›] For ex-
30
CHAPTER 5. GROUP
ample, elements of the fundamental group are represented by loops. The second image at the right shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole. In more recent applications, the influence has also been reversed to motivate geometric constructions by a grouptheoretical background.j[›] In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups.[37] Further branches crucially applying groups include algebraic geometry and number theory.[38] In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups.[39] Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.
Fractions of integers (with b nonzero) are known as rational numbers.l[›] The set of all such fractions is commonly denoted Q.Thereisstillaminorobstaclefor(Q,·), the rationals with multiplication, being a group: because the rational number 0 does not have a multiplicative inverse (i.e., there is no x such that x · 0 = 1), ( Q, ·) is still not a group. However, the set of all nonzero rational numbers Q ∖ {0} = {q ∈ Q | q ≠ 0} does form an abelian group under multiplication, denoted ( Q ∖ {0}, ·).m[›] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of a/b is b/a, therefore the axiom of the inverse element is satisfied. The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and—if division is possible, such as in Q—fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.n[›]
5.5.1
5.5.2
Numbers
Many number systems, such as the integers and the rationals enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups. Integers
The group of integers Z under addition, denoted ( Z, +), has been described above. The integers, with the operation of multiplication instead of addition, (Z, ·) do not form a group. The closure, associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of Z has a (multiplicative) inverse.k[›] Rationals
a . b
Modular arithmetic
0
0
+ 4 h 9
3
6
9
3
6
The hours on a clock form a group that uses addition modulo 12. Here 9 + 4 = 1.
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 “modulo 12” or, in symbols, 9 + 4 ≡ 1 modulo 12.
The desire for the existence of multiplicative inverses suggests considering fractions The group of integers modulo n is written Zn or Z/nZ.
31
5.5. EXAMPLES AND APPLICATIONS
For any prime number p , there is also the multiplicative group of integers modulo p .[40] Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there arefour group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted
z
z
3
2
z
1
0
z =1
16 ≡ 1 (mod 5). The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p 4 5 either, hence the indicated set of classes is closed under z z multiplication. o[›] The identity element is 1, as usual for a multiplicative group, and the associativity follows from 6th complex roots of unity form a cyclic group. z is a primithe corresponding property of integers. Finally, the in- The tive element, but z2 is not, because the odd powers of z are not a verse element axiom requires that given an integer a not power of z2 . divisible by p, there exists an integer b such that a · b ≡ 1 (mod p), i.e. p divides the difference a · b − 1.
The inverse b can be found by using Bézout’s identity and the fact that the greatest common divisor gcd(a, p) equals 1.[41] In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to (Q ∖ {0}, ·) above: it consists of exactly those elements in Z/ pZ that have a multiplicative inverse. [42] These groups are denoted F p× . They arecrucial to publickey cryptography.p[›] 5.5.3
Cyclic groups
Main article: Cyclic group A cyclic group isagroupallofwhoseelementsarepowers of a particular element a .[43] In multiplicative notation, the elements of the group are: ..., a−3 , a−2 , a−1 , a0 = e, a, a2 , a3 , ...,
group. A second example for cyclic groups is the group of n -th complex roots of unity, given by complex numbers z satisfying zn = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°.[44] Using some field theory, the group F p× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 3 1 = 3, 32 = 9 ≡ 4, 3 3 ≡ 2, and 34 ≡ 1. Some cyclic groups have an infinite number of elements. In these groups, for everynon-zero element a,allthepowers of a are distinct; despite the name “cyclic group”, the powers of the elements do not cycle. An infinite cyclic group is isomorphic to ( Z, +), the group of integers under addition introduced above.[45] As these two prototypes are both abelian, so is any cyclic group. The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.[46]
where a2 means a • a, and a−3 stands for a−1 • a−1 • a−1 = (a • a • a)−1 etc.h[›] Such an element a is called a generator or a primitive element of the group. In additive notation, 5.5.4 Symmetry groups therequirement for an element to be primitive is that each element of the group can be written as Main article: Symmetry group See also: Molecular symmetry, Space group, and Symmetry in physics ..., −a−a, −a, 0, a, a+a, ... In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this
Symmetry groups are groups consisting of symmetries of
given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equa-
32 tions and their solutions. [47] Conceptually, group theory can be thought of as the study of symmetry. t[›] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element performs some operation on X compatibly to the group law. In the rightmost example below, an element of order 7ofthe (2,3,7) triangle group actsonthetilingbypermuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.
CHAPTER 5. GROUP
called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition. [51] Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons. Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players.[52] Another application is differentialGalois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. u[›] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.[53] 5.5.5
General linear group and representation theory
Main articles: General linear group and Representation theory Matrix groups consist of matrices together with matrix
Rotations and reflections form the symmetry group of a great icosahedron.
In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. [48] For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved. Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that moleculessometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule.[49][50] Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where thechange from a paraelectricto a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-
Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2.
multiplication. The general linear group GL(n, R ) consists of all invertible n-by-n matrices with real entries.[54] Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.[55] Representation theory is both an application of the group concept and important for a deeper understanding of groups.[56][57] It studies the group by its group actions on other spaces. A broad class of group representations are linear representations, i.e. the group is acting on a vector space, such as the three-dimensional Euclidean space R3 . A representation of G on an n -dimensional real vector space is simply a group homomorphism ρ: G → GL(n, R)
from the group to the general linear group. This way, the
33
5.6. FINITE GROUPS
group operation, which maybe abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.w[›] Given a group action, this gives further meansto studythe object being acted on.x[›] On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.[56][58] 5.5.6
Galois groups
Main article: Galois group Galois groups were developed to help solve polynomial
equations by capturing their symmetry features.[59][60] For example, the solutions of the quadratic equation ax 2 + bx + c = 0 are given by
√ b ± b2 − 4ac − x = . 2a
Exchanging "+" and "−" in the expression, i.e. permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher.[61] Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e. solutions expressible using solely addition, multiplication, and roots similar to the formula above.[62] The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.
5.6 Finite groups Main article: Finite group A group is called finite if it has a finite number of elements. The number of elements is called the order of the group.[63] An important class is the symmetric groups SN , the groups of permutations of N letters. For example, the symmetric group on 3 letters S3 is the group consisting of all possible orderings of the three letters ABC , i.e. contains the elements ABC , ACB, ..., up to CBA, in total 6 (or 3 factorial) elements. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group SN for a suitable integer N (Cayley’s theo-
rem). Parallel to the group of symmetries of the square above, S3 can also be interpreted as the group of symmetries of an equilateral triangle. Theorder of an element a inagroup G is theleast positive integer n such that a n = e, where a n represents
··· a
a,
nfactors
i.e. application of the operation • to n copies of a . (If • represents multiplication, then an corresponds to the nth power of a.) In infinite groups, such an n may not exist, in which case the order of a is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example counting cosets, yield more precise statements about finite groups: Lagrange’s Theorem states that for a finite group G the order of any finite subgroup H divides the order of G . The Sylow theorems give a partial converse. The dihedral group (discussed above) is a finite group of order 8. The order of r1 is 4, as is the order of the subgroup R it generates (see above). The order of the reflection elements fᵥ etc. is 2. Both orders divide 8, as predicted by Lagrange’s theorem. The groups F p× above have order p − 1. 5.6.1
Classification of finite simple groups
Main article: Classification of finite simple groups Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange’s theorem, finite groups of order p , a prime number, are necessarily cyclic (abelian) groups Z p. Groups of order p 2 can also be shown to be abelian, a statement which does not generalize to order p3 , as the non-abelian group D 4 of order 8 = 2 3 above shows.[64] Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.q[›] An intermediate step is the classification of finite simple groups.r[›] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.s[›] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups.[65] Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group—the "monster group"—and certain modular functions, a piece of classical complex analysis,and string theory, a theory supposed to unify the description of many physical phenomena.[66]
34
CHAPTER 5. GROUP
5.7 Groups with additional struc- of real numbers for example: ture Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms)thatmimicthegroupaxioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets. 5.7.1
Topological groups
zw
�
f (x) dx =
f (x + c) dx
for any constant c . Matrix groups over these fields fall under this regime, as do adele rings and adelic algebraic groups, which are basic to number theory. [68] Galois groups of infinite field extensions such as the absolute Galois group can also be equipped with a topology, the so-called Krull topology, which in turn is central to generalize theabove sketched connection of fields andgroups to infinite field extensions.[69] An advanced generalization of this idea, adapted to the needs of algebraic geometry, is the étale fundamental group.[70] 5.7.2
w
�
Lie groups
Main article: Lie group
z
0
1
Lie groups (in honor of Sophus Lie) are groups which also
have a manifold structure, i.e. they are spaces looking locally like some Euclidean space of the appropriate dimension.[71] Again, the additional structure, here the manifold structure, has to be compatible, i.e. the maps corresponding to multiplication and the inverse have to be smooth. A standard example is thegenerallineargroup introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality det (A) ≠ 0,
The unit circle in the complex plane under complex multiplication is a Liegroup and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold andthus a Liegroup, because every smallpiece , such as the red arc in the figure, looks like a part of the real line (shown at the bottom).
Main article: Topological group Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions, that is, g • h, and g−1 must not vary wildly if g and h vary only little. Such groups are called topological groups, and they are the group objects in the category of topological spaces.[67] The most basic examples are the reals R under addition, (R ∖ {0}, ·), and similarly with any other topological field such as the complex numbers or p-adic numbers. All of these groups are locally compact, so they have Haar measures and can be studied via harmonic analysis. The former offer an abstract formalism of invariant integrals. Invariance means, in the case
where A denotes an n-by-n matrix.[72] Lie groups are of fundamental importance in modern physics: Noether’s theorem links continuous symmetries to conserved quantities.[73] Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. v[›] Another example are the Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of space time in special relativity.[74] The full symmetry group of Minkowski space, i.e. including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories.[75] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.[76]
35
5.10. NOTES
5.8
Generalizations
In abstract algebra, more general structures aredefined by relaxing some of the axioms defining a group. [28][77][78] For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers N (including 0) under addition form a monoid, as do the nonzero integers under multiplication (Z ∖ {0}, ·), see above. There is a general method to formally add inverses to elements to any (abelian) monoid, much the same way as ( Q ∖ {0}, ·) is derived from (Z ∖ {0}, ·), known as the Grothendieck group. Groupoids are similar to groups except that the composition a • b need not be defined for all a and b . They arise in the study of more complicated forms of symmetry, often in topological and analytical structures, such as the fundamental groupoid or stacks. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an arbitrary n-ary one (i.e. an operation taking n arguments). With the proper generalization of the group axioms this gives rise to an n -ary group.[79] The table gives a list of several structures generalizing groups.
5.9
See also
• Abelian group • Cyclic group • Euclidean group • Finitely presented group • Free group • Fundamental group • Grothendieck group • Group algebra • Group ring • Heap (mathematics) • List of small groups • Nilpotent group • Non-abelian group • Quantum group • Reductive group • Solvable group • Symmetry in physics • Computational group theory
5.10
Notes
^ a: Mathematical
Reviews lists 3,224 research papers on group theory and its generalizations written in 2005. ^ aa: The classification was announced in 1983, but gaps were found in the proof. See classification of finite simple groups for further information. ^ b: The closure axiom is already implied by the condition that • be a binary operation. Some authors therefore omit this axiom. However, group constructions often start with an operation defined on a superset, so a closure step is common in proofs that a system is a group. Lang 2002 ^ c: See, for example, the books of Lang (2002, 2005) and Herstein (1996, 1975). ^ d: However, a group is not determined by its lattice of subgroups. See Suzuki 1951. ^ e: The fact that the group operation extends this canonically is an instance of a universal property. ^ f: For example, if G is finite, then the size of any subgroup and any quotient group divides the size of G , according to Lagrange’s theorem. ^ g: The word homomorphism derives from Greek ὁμός—the same and μορφή—structure. ^ h: The additive notation for elements of a cyclic group would be t • a, t in Z. ^ i: See the Seifert–van Kampen theorem for an example. ^ j: An example is group cohomology of a group which equals the singular cohomology of its classifying space. ^ k: Elements which do have multiplicative inverses are called units, see Lang 2002, §II.1, p. 84. ^ l: The transition from the integers to the rationals by adding fractions is generalized by the quotient field. ^ m: The same is true for any field F instead of Q . See Lang 2005, §III.1, p. 86. ^ n: For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See Lang 2002, Theorem IV.1.9. The notions of torsion of a module and simple algebras are other instances of this principle. ^ o: The stated property is a possible definition of prime numbers. See prime element. ^ p: For example, the Diffie-Hellman protocol uses the discrete logarithm. ^ q: The groups of order at most 2000 are known. Up to isomorphism, there are about 49 billion. See Besche, Eick & O'Brien 2001. ^ r: The gap between the classification of simple groups and the one of all groups lies in the extension problem, a problem too hard to be solved in general. See Aschbacher 2004, p. 737. ^ s: Equivalently, a nontrivial group is simple if its only quotient groups are the trivial group and the group itself. See Michler 2006, Carter 1989. ^ t: More rigorously, every group is the symmetry group of some graph; see Frucht’s theorem, Frucht 1939. ^ u: More precisely, the monodromy action on the vector space of solutions of the differential equations is
36
CHAPTER 5. GROUP
considered. See Kuga 1993, pp. 105–113. ^ v: See Schwarzschild metric for an example where symmetry greatly reduces the complexity of physical systems. ^ w: This was crucial to the classification of finite simple groups, for example. See Aschbacher 2004. ^ x: See, for example, Schur’s Lemma for the impact of a group action on simple modules. A more involved example is the action of an absolute Galois group on étale cohomology. ^ y: Injective and surjective maps correspond to monoand epimorphisms, respectively. They are interchanged when passing to the dual category.
[24] Ledermann 1953, §1.2, pp. 4–5 [25] Ledermann 1973, §I.1, p. 3 [26] Lang 2002, §I.2, p. 7 [27] Lang 2005, §II.1, p. 17 [28] Mac Lane 1998 [29] Lang 2005, §II.3, p. 34 [30] Lang 2005, §II.1, p. 19 [31] Ledermann 1973, §II.12, p. 39 [32] Lang 2005, §II.4, p. 41 [33] Lang 2002, §I.2, p. 12
5.11
Citations
[1] Herstein 1975, §2, p. 26
[34] Lang 2005, §II.4, p. 45 [35] Lang 2002, §I.2, p. 9 [36] Hatcher 2002, Chapter I, p. 30
[2] Hall 1967, §1.1, p. 1: “The idea of a group is one which [37] Coornaert, Delzant & Papadopoulos 1990 pervades the whole of mathematics both pure and ap[38] for example, class groups and Picard groups; see Neukirch plied.” 1999, in particular §§I.12 and I.13 [3] Lang 2005, App. 2, p. 360 [39] Seress 1997 [4] Cook, Mariana R. (2009), Mathematicians: An Outer View of the Inner World , Princeton, N.J.: Princeton Uni- [40] Lang 2005, Chapter VII versity Press, p. 24, ISBN 9780691139517 [41] Rosen 2000, p. 54 (Theorem 2.1) [5] Herstein 1975, §2.1, p. 27 [42] Lang 2005, §VIII.1, p. 292 [6] Weisstein, Eric W., “Identity Element”, MathWorld .
[43] Lang 2005, §II.1, p. 22
[7] Herstein 1975, §2.6, p. 54
[44] Lang 2005, §II.2, p. 26
[8] Wussing 2007
[45] Lang 2005, §II.1, p. 22 (example 11)
[9] Kleiner 1986
[46] Lang 2002, §I.5, p. 26, 29
[10] Smith 1906
[47] Weyl 1952
[11] Galois 1908
[48] Conway, Delgado Friedrichs & Huson et al. 2001. See also Bishop 1993
[12] Kleiner 1986, p. 202 [13] Cayley 1889
[49] Bersuker, Isaac (2006), The Jahn-Teller Effect , Cambridge University Press, p. 2, ISBN 0-521-82212-2
[14] Wussing 2007, §III.2
[50] Jahn & Teller 1937
[15] Lie 1973 [16] Kleiner 1986, p. 204
[51] Dove, Martin T (2003), Structure and Dynamics: an atomic view of materials, Oxford University Press, p. 265, ISBN 0-19-850678-3
[17] Wussing 2007, §I.3.4
[52] Welsh 1989
[18] Jordan 1870
[53] Mumford, Fogarty & Kirwan 1994
[19] von Dyck 1882
[54] Lay 2003
[20] Curtis 2003
[55] Kuipers 1999
[21] Mackey 1976
[56] Fulton & Harris 1991
[22] Borel 2001
[57] Serre 1977
[23] Aschbacher 2004
[58] Rudin 1990
37
5.12. REFERENCES [59] Robinson 1996, p. viii [60] Artin 1998 [61] Lang 2002, Chapter VI (see in particular p. 273 for concrete examples) [62] Lang 2002, p. 292 (Theorem VI.7.2) [63] Kurzweil & Stellmacher 2004 [64] Artin 1991, Theorem 6.1.14. See also Lang 2002, p. 77 for similar results. [65] Lang 2002, §I. 3, p. 22 [66] Ronan 2007 [67] Husain 1966 [68] Neukirch 1999 [69] Shatz 1972 [70] Milne 1980 [71] Warner 1983 [72] Borel 1991 [73] Goldstein 1980 [74] Weinberg 1972
• Herstein,
Israel Nathan (1996), Abstract algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., ISBN 978-0-13-374562-7, MR 1375019.
• Herstein, Israel Nathan (1975), Topics in algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, MR 0356988.
• Lang, Serge (2002), Algebra, Graduate Texts in
Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3.
• Ledermann, Walter (1953), Introduction to the the-
ory of finite groups, Oliver and Boyd, Edinburgh and
London, MR 0054593.
• Ledermann, Walter (1973), Introduction to group
theory, New York: Barnes and Noble, OCLC
795613.
• Robinson, Derek John Scott (1996), A course in the theory of groups , Berlin, New York: Springer-
Verlag, ISBN 978-0-387-94461-6.
[75] Naber 2003 [76] Becchi 1997 [77] Denecke & Wismath 2002 [78] Romanowska & Smith 2002 [79] Dudek 2001
5.12 References 5.12.1 General references
• Artin,
Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.
• Devlin, Keith (2000), The Language of Mathemat-
ics: Making the Invisible Visible , Owl Books, ISBN
978-0-8050-7254-9, Chapter 5 provides a laymanaccessible explanation of groups.
• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathe-
matics, Readings in Mathematics 129, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249, ISBN 978-0-387-97527-6.
• Hall, G. G. (1967), Applied group theory , American Elsevier Publishing Co., Inc., New York, MR 0219593, an elementary introduction.
5.12.2 Special references
• Artin,
Emil (1998), Galois Theory , New York: Dover Publications, ISBN 978-0-486-62342-9.
• Aschbacher, Michael (2004), “The Status of the
Classification of the Finite Simple Groups” (PDF), Notices ofthe American Mathematical Society 51 (7): 736–740.
• Becchi,
C. (1997), Introduction to Gauge 5211, arXiv:hep-ph/9705211, Bibcode:1997hep.ph....5211B. Theories, p.
• Besche, Hans Ulrich; Eick, Bettina; O'Brien, E.
A. (2001), “The groups of order at most 2000”,
Electronic Research Announcements of the American Mathematical Society 7: 1–4, doi:10.1090/S1079-
6762-01-00087-7, MR 1826989.
• Bishop, David H. L. (1993), Group theory and chemistry, New York: Dover Publications, ISBN
978-0-486-67355-4.
• Borel,
Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387-97370-8, MR 1102012.
• Carter, Roger W. (1989), Simple groups of Lie type , New York: John Wiley & Sons, ISBN 978-0-47150683-6.
38
CHAPTER 5. GROUP
• Conway, John Horton; Delgado Friedrichs, Olaf;
Huson, Daniel H.; Thurston, William P. (2001), “On three-dimensional space groups”, Beiträge zur Algebra und Geometrie 42 (2): 475–507, arXiv:math.MG/9911185, MR 1865535.
• Coornaert,
• Mac
Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2.
• Michler, Gerhard (2006), Theory of finite simple
groups, Cambridge University Press, ISBN 978-0-
M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes [Geometry and Group Theory] , Lecture Notes in Mathematics (in French) 1441, Berlin, New York: SpringerVerlag, ISBN 978-3-540-52977-4, MR 1075994.
• Milne, James S. (1980), Étale cohomology, Prince-
• Denecke, Klaus; Wismath, Shelly L. (2002), Uni-
Geometric invariant theory 34 (3rd ed.), Berlin, New
versal algebra and applications in theoretical com puter science, London: CRC Press, ISBN 978-1-
58488-254-1.
• Dudek, W.A. (2001), “On some old problems in
n-ary groups”, Quasigroups and Related Systems 8 : 15–36.
• Frucht, R. (1939), “Herstellung von Graphen mit
vorgegebener abstrakter Gruppe [Construction of Graphs with Prescribed Group]", Compositio Mathematica (in German) 6: 239–50.
• Goldstein,
Herbert (1980), Classical Mechanics (2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588–596, ISBN 0-201-02918-9.
• Hatcher,
Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-52179540-1.
• Husain, Taqdir (1966), Introduction to Topological Groups, Philadelphia: W.B. Saunders Company,
ISBN 978-0-89874-193-3
• Jahn,
H.; Teller, E. (1937), “Stability of Polyatomic Molecules in Degenerate Electronic States. I. Orbital Degeneracy”, Proceedings of the Royal Society A 161 (905): 220–235, Bibcode:1937RSPSA.161..220J, doi:10.1098/rspa.1937.0142.
• Kuipers, Jack B. (1999), Quaternions and rota-
tion sequences—A primer with applications to orbits, aerospace, and virtual reality , Princeton University
Press, ISBN 978-0-691-05872-6, MR 1670862.
• Kuga, Michio (1993), Galois’ dream: group theory
and differential equations , Boston, MA: Birkhäuser
Boston, ISBN 978-0-8176-3688-3, MR 1199112.
• Kurzweil, Hans; Stellmacher, Bernd (2004),
The theory of finite groups , Universitext, Berlin, New
York: Springer-Verlag, ISBN 978-0-387-40510-0, MR 2014408.
• Lay, David (2003), Linear Algebra and Its Applications, Addison-Wesley, ISBN 978-0-201-70970-4.
521-86625-5.
ton University Press, ISBN 978-0-691-08238-7
• Mumford, David; Fogarty, J.; Kirwan, F. (1994), York: Springer-Verlag, ISBN 978-3-540-56963-3, MR 1304906.
• Naber,
Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR 2044239.
• Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften 322, Berlin: Springer-Verlag, ISBN 978-3-54065399-8, Zbl 0956.11021, MR 1697859.
• Romanowska, A.B.; Smith, J.D.H. (2002), Modes, World Scientific, ISBN 978-981-02-4942-7.
• Ronan, Mark (2007), Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, Oxford University Press, ISBN 978-0-19-
280723-6.
• Rosen, Kenneth H. (2000), Elementary number theory and its applications (4th ed.), Addison-Wesley,
ISBN 978-0-201-87073-2, MR 1739433.
• Rudin, Walter (1990), Fourier Analysis on Groups , Wiley Classics, Wiley-Blackwell, ISBN 0-47152364-X.
• Seress, Ákos (1997), “An introduction to computational group theory”, Notices of the American Mathematical Society 44 (6): 671–679, MR 1452069.
• Serre, Jean-Pierre (1977), Linear representations of finite groups, Berlin, New York: Springer-Verlag,
ISBN 978-0-387-90190-9, MR 0450380.
• Shatz, Stephen S. (1972), Profinite groups, arith-
metic, and geometry, Princeton University Press,
ISBN 978-0-691-08017-8, MR 0347778
• Suzuki, Michio (1951), “On the lattice of sub-
groups of finite groups”, Transactions of the American Mathematical Society 70 (2): 345–371, doi:10.2307/1990375, JSTOR 1990375.
• Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups , Berlin, New York:
Springer-Verlag, ISBN 978-0-387-90894-6.
39
5.12. REFERENCES
• Weinberg, Steven (1972), Gravitation and Cosmol-
ogy, New York: John Wiley & Sons, ISBN 0-471-
92567-5.
• Welsh, Dominic (1989), Codes and cryptography , Oxford: Clarendon Press, ISBN 978-0-19-8532873.
• Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 978-0-691-02374-8.
5.12.3 Historical references
See also: Historically important publications in group theory
• Borel, Armand (2001), Essays in the History of Lie Groups and Algebraic Groups , Providence,
R.I.: American Mathematical Society, ISBN 9780-8218-0288-5
• Cayley, Arthur (1889), The collected mathemat-
ical papers of Arthur Cayley , II (1851–1860),
Cambridge University Press.
• O'Connor, J.J; Robertson, E.F. (1996), The development of group theory .
• Curtis, Charles W. (2003), Pioneers of Represen-
tation Theory: Frobenius, Burnside, Schur, and Brauer , History of Mathematics, Providence, R.I.:
American Mathematical Society, ISBN 978-08218-2677-5.
• von Dyck, Walther (1882), “Gruppentheoretische Studien
(Group-theoretical
Studies)",
Mathematische Annalen (in German) 20 (1):
1–44, doi:10.1007/BF01443322.
• Galois,
Évariste (1908), Tannery, Jules, ed.,
Manuscrits de Évariste Galois [Évariste Galois’ Manuscripts] (in French), Paris: Gauthier-Villars
(Galois work was first published by Joseph Liouville in 1843).
• Jordan, Camille (1870), Traité des substitutions et des équations algébriques [Study of Substitutions and Algebraic Equations] (in French), Paris: Gauthier-
Villars.
• Kleiner, Israel (1986), “The evolution of group the-
ory: a brief survey”, Mathematics Magazine 59 (4): 195–215, doi:10.2307/2690312, MR 863090.
• Lie,
Sophus (1973), Gesammelte Abhandlungen. Band 1 [Collected papers. Volume 1] (in German), New York: Johnson Reprint Corp., MR 0392459.
• Mackey, George Whitelaw (1976), The theory of unitary group representations , University of Chicago
Press, MR 0396826
• Smith, David Eugene (1906), History of Modern Mathematics, Mathematical Monographs, No. 1.
• Wussing, Hans (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory , New York: Dover
Publications, ISBN 978-0-486-45868-7.
Chapter 6
Group theory This article covers advanced notions. For basic topics, see Group (mathematics). For group theory in social sciences, see social group. In mathematics and abstract algebra, group theory
One of the most important mathematical achievements of the 20th century[1] was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 1980, that culminated in a complete classification of finite simple groups.
6.1
Main classes of groups
Main articles: Group (mathematics) and Glossary of group theory The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstract groups that may be specified through a presentation by generators and relations. 6.1.1
The popular puzzle Rubik’s cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation groups.
studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Liegroups aretwobranches of group theorythat have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also central to public key cryptography.
Permutation groups
Thefirst classof groupsto undergo a systematicstudywas permutation groups. Given any set X and a collection G of bijections of X into itself (known as permutations ) that is closed under compositions and inverses, G is a group acting on X . If X consists of n elements and G consists of all permutations, G isthe symmetric group Sn; in general, any permutation group G is a subgroup of the symmetric group of X . An earlyconstruction dueto Cayley exhibited any group as a permutation group, acting on itself ( X = G ) by means of the left regular representation. In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for n ≥ 5, the alternating group An is simple, i.e. does not admit any proper normal subgroups. This fact plays a key role in the impossibility of solving a general algebraic equation of degree n' ≥ 5 in radicals . 6.1.2
Matrix groups
The next important class of groups is given by matrix groups, or linear groups. Here G is a set consisting of invertible matrices of given order n over a field K that is
40
41
6.2. BRANCHES OF GROUP THEORY
closed under the products and inverses. Such a group acts on the n-dimensional vector space K n by linear transformations. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group G . 6.1.3
Transformation groups
importance for the development of mathematics: it foreshadowed the creation of abstract algebra in the works of Hilbert, Emil Artin, Emmy Noether, and mathematicians of their school. 6.1.5
Topological and algebraic groups
An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of Permutation groups and matrix groups are special cases a topological space, differentiable manifold, or algebraic of transformation groups: groups that act on a certain variety. If the group operations m (multiplication) and i space X preserving its inherent structure. In the case of (inversion), permutation groups, X is a set; for matrix groups, X is a vector space. The concept of a transformation group is closely related with the concept of a symmetry group: m : G × G → G, (g, h) → gh, i : G → G, g → g −1 , transformation groups frequently consist of all transformations that preserve a certain structure. are compatible with this structure, i.e. are continuous, The theory of transformation groups forms a bridge con- smooth or regular (in the sense of algebraic geometry) necting group theory with differential geometry. A long maps, then G becomes[2]a topological group, a Lie group, line of research, originating with Lie and Klein, consid- or an algebraic group. ers group actions on manifolds by homeomorphisms or The presence of extra structure relates these types of diffeomorphisms. The groupsthemselves may be discrete groups with other mathematical disciplines and means or continuous. that more tools are available in their study. Topological groups form a natural domain for abstract harmonic analysis, whereas Lie groups (frequently realized as transfor6.1.4 Abstract groups mation groups) are the mainstays of differential geometry and unitary representation theory. Certain classificaMost groups considered in the first stage of the develop- tion questions that cannot be solved in general can be apment of group theory were “concrete”, having been real- proached and resolved for special subclasses of groups. ized through numbers, permutations, or matrices. It was Thus, compact connected Lie groups have been comnot until the late nineteenth century that the idea of an pletely classified. There is a fruitful relation between inabstract group as a set with operations satisfying a certain finite abstract groups and topological groups: whenever system of axioms began to take hold. A typical way of a group Γ can be realized as a lattice in a topological specifying an abstract group is through a presentation by group G , the geometry and analysis pertaining to G yield generators and relations , important results about Γ . A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for G = ⟨S |R⟩. example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and A significant source of abstract groups is given by the properties of G translate into the properties of its finite construction of a factor group , or quotient group, G /H , quotients. of a group G by a normal subgroup H . Class groups of algebraic number fields were among the earliest examples of factor groups, of much interest in number theory. If a group G is a permutation group on a set X , the fac- 6.2 Branches of group theory tor group G /H is no longer acting on X ; but the idea of an abstract group permits one not to worry about this dis- 6.2.1 Finite group theory crepancy. The change of perspective from concrete to abstract Main article: Finite group groups makes it natural to consider properties of groups that areindependentof a particular realization, or in mod- During the twentieth century, mathematicians investiern language, invariant under isomorphism, as well as the gated some aspects of the theory of finite groups in great classes of group with a given such property: finite groups, depth, especially the local theory of finite groups and periodic groups, simple groups, solvable groups, and so the theory of solvable and nilpotent groups. As a conseon. Rather than exploring properties of an individual quence, the complete classification of finite simplegroups group, one seeks to establish results that apply to a whole was achieved, meaning that all those simple groups from class of groups. The new paradigm was of paramount which all finite groups can be built are now known.
42
CHAPTER 6. GROUP THEORY
During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite groups often occur when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of Lie groups, which may be viewed as dealing with "continuous symmetry", is strongly influenced by the associated Weyl groups. These are finite groups generated by reflections which act on a finitedimensional Euclidean space. The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.
6.2.3
Lie theory
Main article: Lie group
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, page 3.[5] Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of 6.2.2 Representation of groups differentialequations (differentialGalois theory),inmuch the same way as permutation groups are used in Galois Main article: Representation theory theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of Saying that a group G acts on a set X means that every continuous symmetry groups was one of Lie’s principal element of G defines a bijective map on the set X in a motivations. way compatible with the group structure. When X has more structure, it is useful to restrict this notion further: a representation of G on a vector space V is a group ho- 6.2.4 Combinatorial and geometric group momorphism: theory ρ : G → GL(V ),
where GL(V ) consists of the invertible linear transformations of V . In other words, to every group element g is assigned an automorphism ρ( g) such that ρ( g) ∘ ρ(h) = ρ( gh) for any h in G . This definition can be understood in two directions, both of which give rise to whole new domains of mathematics. [3] On the one hand, it may yield new information about the group G : often, the group operation in G is abstractly given, but via ρ , it corresponds to the multiplication of matrices, which is very explicit. [4] On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if G is finite, it is known that V above decomposes into irreducible parts. These parts in turn are much more easily manageable than the whole V (via Schur’s lemma). Given a group G , representation theory then asks what representations of G exist. There are several settings, and theemployed methods andobtainedresults arerather different in every case: representation theory of finite groups and representations of Lie groups are two main subdomains of the theory. The totality of representations is governed by the group’s characters. For example, Fourier polynomials can be interpreted as the characters of U(1), the group of complex numbers of absolute value 1, acting on the L2-space of periodic functions.
Main article: Geometric group theory Groups can be described in different ways. Finite groups can be described by writing down the group table consisting of all possible multiplications g • h. A more compact way of defining a group is by generators and relations , also called the presentation of a group. Given any set F of generators{ gi }i ∈ I , the free group generated by F subjects onto the group G . The kernel of this map is called subgroup of relations, generated by some subset D . The presentation is usually denoted by F | D . For example, the group Z = a | can be generated by one element a (equal to +1 or −1) and no relations, because n · 1 never equals 0 unless n is zero. A string consisting of generator symbols and their inverses is called a word . Combinatorial group theory studies groups from the perspective of generators and relations.[6] It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of graphs via their fundamental groups. For example, one can show that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the problem to Turing machines, one can show that thereisingeneralno algorithmsolving this task. Another,
43
6.4. APPLICATIONS OF GROUP THEORY
generally harder, algorithmically insoluble problem is the group isomorphism problem, which asks whether two groups given by different presentations are actually isomorphic. For example, the additive group Z of integers can also be presented by x , y | xyxyx = e; it may not be obvious that these groups are isomorphic.[7]
2. If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry). The corresponding group is called isometry group of X . 3. If instead angles are preserved, one speaks of conformal maps. Conformal maps give rise to Kleinian groups, for example. 4. Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation
b x2
e
Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.[8] The first idea is made precise by means of the Cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the word metric given by the length of the minimal path between the elements. A theorem of Milnor and Svarc then says that given a group G acting in a reasonable manner on a metric space X , for example a compact manifold, then G is quasi-isometric (i.e. looks similar from a distance) to the space X .
√
The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a symmetry of an object, and then apply another symmetry, theresult will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions are associative. Frucht’s theorem says that every group is the symmetry group of some graph. So every abstract group is actually the symmetries of some explicit object. The saying of “preserving the structure” of an object can be made precise by working in a category. Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.
6.4 6.3
√
has the two solutions + 3 , and − 3 . In this case, the group that exchanges the two roots is the Galois group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.
a
The Cayley graph of x, y ∣ , the free group of rank 2.
−3 = 0
Applications of group theory
Connection of groups and sym- Applications of group theory abound. Almost all structures in abstract algebra are special cases of groups. metry
Rings, for example, can be viewed as abelian groups (corresponding to addition) together with a second operation Main article: Symmetry group (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of Given a structured object X of any sort, a symmetry is those entities. a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example 6.4.1 Galois theory 1. If X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise Main article: Galois theory to permutation groups.
44
CHAPTER 6. GROUP THEORY
Galois theory uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The fundamental theorem of Galois theory provides a link between algebraic field extensions and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding Galois group. For example, S 5 , the symmetric group in 5 elements, is not solvable which implies that The cyclic group Z 26 underlies Caesar’s cipher . the general quintic equation cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still 6.4.3 Algebraic geometry and cryptografruitfully applied to yield new resultsin areas such as class phy field theory. Main articles: Algebraic geometry and Cryptography
6.4.2
Algebraic topology
Main article: Algebraic topology Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called “invariants” because they are defined in such a way that they do not change if the space is subjected to some deformation. For example, the fundamental group “counts” how many paths in the space are essentially different. The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory.
Algebraic geometry and cryptography likewise uses group theory in many ways. Abelian varieties have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures.[9] The one-dimensional case, namely elliptic curves is studied in particular detail. They are both theoretically and practically intriguing.[10] Very large groups of prime order constructed in Elliptic-Curve Cryptography serve for public key cryptography. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the discrete logarithm very hard to calculate. One of the earliest encryption protocols, Caesar’s cipher, may also be interpreted as a (very easy) group operation. In another direction, toric varieties are algebraic varieties acted on by a torus. Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.[11]
6.4.4
Algebraic number theory
Main article: Algebraic number theory Algebraic number theory isaspecialcaseofgrouptheory, thereby following the rules of the latter. For example, Euler’s product formula 1
� �
n 1
≥
ns
=
pprime
1 1
s
− p−
captures the fact that any integer decomposes in a unique A torus. Its abelian group structure is induced from the map C → way into primes. The failure of this statement for more C / Z + τZ , where τ is a parameter living in the upper half plane. general rings gives rise to class groups and regular primes, which feature in Kummer’s treatment of Fermat’s Last Theorem.
45
6.4. APPLICATIONS OF GROUP THEORY
6.4.5
Harmonic analysis
Main article: Harmonic analysis Analysis on Lie groups and certain other groups is called harmonic analysis. Haar measures, that is, integrals invariant under the translation in a Lie group, are used for pattern recognition and other image processing techniques. [12]
the StandardModel, gauge theory,the Lorentz group,and the Poincaré group. 6.4.9
Chemistry and materials science
In chemistry and materials science, groups are used to classify crystal structures, regular polyhedra, and the symmetries of molecules. The assigned point groups can then be used to determine physical properties (such as polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spec6.4.6 Combinatorics troscopy, circular dichroism spectroscopy, magnetic cirIn combinatorics, the notion of permutation group and cular dichroism spectroscopy, UV/Vis spectroscopy, and the concept of group action are often used to simplify the fluorescence spectroscopy), and to construct molecular counting of a set of objects; see in particular Burnside’s orbitals. lemma. Molecular symmetry is responsible for many physical and spectroscopic properties of compounds and provides relevant information about howchemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called “symmetry elements”. These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule.
The circle of fifths may be endowed with a cyclic group structure
6.4.7
Music
The presence of the 12-periodicity in the circle of fifths yields applications of elementary group theory in musical set theory. 6.4.8
Physics
In physics, groups are important because they describe the symmetries which the laws of physics seem to obey. According to Noether’s theorem, every continuous symmetry of a physical system corresponds to a conservation law of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the “possible” physical theories. Examples of theuseof groupsin physics include
Water molecule with symmetry axis
In chemistry, there are five important symmetry operations. The identity operation (E) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a chiral molecule consists of only the identity operation. Rota-
46
CHAPTER 6. GROUP THEORY
tion around an axis (C n) consists of rotating the molecule around a specific axis by a specific angle. For example, if a water molecule rotates 180° around the axis that passes through the oxygen atom and between the hydrogen atoms, it is in the same configuration as it started. In this case, n = 2, since applying it twice produces the identity operation. Other symmetry operations are: reflection, inversion and improper rotation (rotation followed by reflection).[13] 6.4.10 Statistical Mechanics
classification of finite simple groups isavastbodyofwork from themid 20th century, classifying allthe finite simple groups.
6.6
• Glossary of group theory • List of group theory topics
Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed 6.7 by Willard Gibbs, relating to the summing of an infinite number of probabilities to yield a meaningful solution [14] [1]
6.5
History
Main article: History of group theory Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. The number-theoretic strand was begun by Leonhard Euler, and developed by Gauss’s work on modular arithmetic and additive and multiplicative groups related to quadratic fields. Early results about permutation groups were obtained by Lagrange, Ruffini, and Abel in their quest for general solutions of polynomial equations of high degree. Évariste Galois coined the term “group” and established a connection, now known as Galois theory, between the nascent theory of groups and field theory. In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations. Arthur Cayley and Augustin Louis Cauchy pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from geometrical situations. In an attempt to come to grips with possible geometries (such as euclidean, hyperbolic or projective geometry) using group theory, Felix Klein initiated the Erlangen programme. Sophus Lie, in 1884, started using groups(now called Lie groups) attached to analytic problems. Thirdly, groups were, at first implicitly and later explicitly, used in algebraic number theory. The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around1880. Sincethen, theimpact of group theory has been ever growing, giving rise to the birth of abstract algebra in the early 20th century, representation theory, and many more influential spin-off domains. The
See also
Notes
• Elwes, Richard, "An enormous theorem: the classification of finite simple groups," Plus Magazine, Issue 41, December 2006.
[2] This process of imposing extra structure has been formalized through the notion of a group object in a suitable category. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties. [3] Such as group cohomology or equivariant K-theory. [4] In particular, if the representation is faithful. [5] Arthur Tresse (1893). “Sur les invariants différentiels des groupes continus de transformations”. Acta Mathematica 18: 1–88. doi:10.1007/bf02418270. [6] Schupp & Lyndon 2001 [7] Writing z = xy, one has G = z, y | z3 = y = z. [8] La Harpe 2000 [9] For example the Hodge conjecture (in certain cases). [10] See the Birch-Swinnerton-Dyer conjecture, one of the millennium problems [11] Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), “Torification and factorization of birational maps”, Journal of the American Mathematical Society 15 (3): 531–572, doi:10.1090/S0894-0347-0200396-X, MR 1896232 [12] Lenz, Reiner (1990), Group theoretical methods in image processing, Lecture Notes in Computer Science 413 , Berlin, New York: Springer-Verlag, doi:10.1007/3-54052290-5, ISBN 978-0-387-52290-6 [13] Shriver, D.F.; Atkins, P.W. Química Inorgânica, 3ª ed., Porto Alegre, Bookman, 2003. [14] Norber Weiner, Cybernetics: Or Control and Communication in the Animal and the Machine, Ch 2
6.9. EXTERNAL LINKS
47
6.8 References
• Schupp, Paul E.; Lyndon, Roger C. (2001), Combi-
• Borel,
Armand (1991), Linear algebraic groups , Graduate Texts in Mathematics 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387-97370-8, MR 1102012
• Carter,
Nathan C. (2009), Visual group theory, Classroom Resource Materials Series, Mathematical Association of America, ISBN 978-0-88385-757-1, MR 2504193
• Cannon,
John J. (1969), “Computers in group theory: A survey”, Communications of the Association for Computing Machinery 12: 3–12, doi:10.1145/362835.362837, MR 0290613
• Frucht, R. (1939), “Herstellung von Graphen mit
vorgegebener abstrakter Gruppe”, Compositio Mathematica 6: 239–50, ISSN 0010-437X
natorial group theory , Berlin, New York: Springer-
Verlag, ISBN 978-3-540-41158-1
• Scott, W. R. (1987) [1964], Group Theory, New
York: Dover, ISBN 0-486-65377-3 Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.
• Shatz, Stephen S. (1972), Profinite groups, arith-
metic, and geometry, Princeton University Press,
ISBN 978-0-691-08017-8, MR 0347778
• Weibel, Charles A. (1994), An introduction to ho-
mological algebra, Cambridge Studies in Advanced
Mathematics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324
Stewart, Ian (2006), “Non- 6.9 External links linear dynamics of networks: the groupoid for• History of the abstract group concept malism”, Bull. Amer. Math. Soc. (N.S.) 43 (03): 305–364, doi:10.1090/S0273-0979-06• Higher dimensional group theory This presents a 01108-6, MR2223010 Shows theadvantage of genview of group theory as level one of a theory which eralising from group to groupoid. extends in alldimensions, andhasapplications in ho• Judson, Thomas W. (1997), Abstract Algebra: Themotopy theory and to higherdimensionalnonabelian ory and Applications An introductory undergradumethods for local-to-global problems. ate text in the spirit of texts by Gallian or Herstein, • Plus teacher and student package: Group Theory covering groups, rings, integral domains, fields and This package bringstogether allthearticles on group Galois theory. Free downloadable PDF with opentheory from Plus, the online mathematics magazine source GFDL license. produced by the Millennium Mathematics Project at • Kleiner, Israel (1986), “The evolution of group thethe University of Cambridge, exploring applications ory: a brief survey”, Mathematics Magazine 59 (4): and recent breakthroughs, and giving explicit defi195–215, doi:10.2307/2690312, ISSN 0025-570X, nitions and examples of groups. JSTOR 2690312, MR 863090 • US Naval Academy group theory guide A general • La Harpe, Pierre de (2000), Topics in geometric introduction to group theory with exercises written group theory, University of Chicago Press, ISBN by Tony Gaglione. 978-0-226-31721-2
• Golubitsky, Martin;
• Livio, M. (2005), The Equation That Couldn't Be
Solved: How Mathematical Genius Discovered the Language of Symmetry, Simon & Schuster, ISBN 0-
7432-5820-7 Conveys the practical value of group theory by explaining how it points to symmetries in physics and other sciences.
• Mumford, David (1970), Abelian varieties , Oxford
University Press, ISBN 978-0-19-560528-0, OCLC 138290
• Ronan M., 2006. Symmetry and the Monster . Ox-
ford University Press. ISBN 0-19-280722-6. For lay readers. Describes the quest to find the basic building blocks for finite groups.
• Rotman, Joseph (1994), An introduction to the the-
ory of groups , New York: Springer-Verlag, ISBN 0-
387-94285-8 A standard contemporary reference.
Chapter 7
Homomorphism Not to be confused with holomorphism homeomorphism.
or
• A module homomorphism is a map that preserves module structures.
• An algebra homomorphism is a homomorphism that In abstract algebra, a homomorphism is a structurepreserves the algebra structure. preserving map between two algebraic structures (such • A functor is a homomorphism between two as groups, rings, or vector spaces). The word homocategories. morphism comes from the ancient Greek language: ὁμός (homos) meaning “same” and μορφή (morphe) meaning “form” or “shape”. Isomorphisms, automorphisms, and Not all structure that an object possesses need be preendomorphisms are special types of homomorphisms. served by a homomorphism. For example, one may have a semigroup homomorphism between two monoids, and this will not be a monoid homomorphism if it does not map the identity of the domain to that of the codomain. 7.1 Definition and illustration The algebraic structure to be preserved may include more than one operation, and a homomorphism is required to 7.1.1 Definition preserve each operation. For example, a ring has both A homomorphism is a map that preserves selected struc- addition and multiplication, and a homomorphism from ture between two algebraic structures, with the structure the ring (R, +, ∗, 0, 1) to the ring ( R′, +′, ∗′, 0′, 1′) is to be preserved being given by the naming of the homo- a function such that f (r + s ) = f (r ) +′ f (s), f (r ∗ s) = f (r ) ∗′ f (s) and f (1) = 1′ for any elements r and s of the morphism. Particular definitions of homomorphism include the fol- domain ring. If rings are not required to be unital, thelast condition is omitted. In addition, if defining structures of lowing: (e.g. 0 and additive inverses in the case of a ring) were necessarily preserved by the above, preserving these • A semigroup homomorphism isamapthatpreserves not would be added requirements. an associative binary operation. notion of a homomorphism can be given a formal • A monoid homomorphism is a semigroup homo- The morphism that maps theidentityelement to theiden- definition in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this tity of the codomain. setting, a homomorphism f : A → B is a function between • A group homomorphism is a homomorphism that two algebraic structures of the same type such that preserves the group structure. It may equivalently be defined as a semigroup homomorphism between f (μA(a1 , ..., an)) = μB( f (a1 ), ..., f (an)) groups. • A ring homomorphism is a homomorphism that pre- for each n-ary operation μ and for all elements a1, ..., an serves the ring structure. Whether the multiplicative ∈ A. identity is to be preserved depends upon the defini- The function between two algebraic structures of the tion of ring in use. same type is a reduction of the structure group. H to G • A linear map is a homomorphism that preserves is also called the G-structure. For example, a group is an the vector space structure, namely the abelian group algebraic object consisting of a set together with a single structure and scalar multiplication. The scalar type binary operation, satisfying certain axioms. If ( G , ∗) and must further be specified to specify the homomor- (H , ∗′) are groups, a homomorphism from (G , ∗) to (H , phism, e.g. every R -linear map is a Z -linear map, ∗′) is a function f : (G , ∗) → (H , ∗′) such that f ( g1 ∗ g2 ) = but not vice versa. f ( g1 ) ∗′ f ( g2 ) for all elements g1 , g 2 ∈ G . Since inverses 48
49
7.2. INFORMAL DISCUSSION
exist in G and H , onecanshowthattheidentityof G maps That is, ƒ ( z) is the absolute value (or modulus) of the to the identity of H and that inverses are preserved. complex number z. Then f is a homomorphism of groups, since it preserves multiplication: 7.1.2
Basic examples
f ( z1 z 2 ) = |z1 z 2 | = |z1 | |z2 | = f( z1 ) f( z2 ).
Note that ƒ cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition: |z1 + z2 | ≠ |z1 | + |z2 |.
As another example, the picture shows a monoid homomorphism f from the monoid (N, +, 0) to the monoid (N, ×, 1). Due to the different names of corresponding operations, the structure preservation properties satisfied by f amount to f ( x + y) = f ( x ) × f (y) and f (0) = 1.
Monoid homomorphism f from the monoid ( N , +, 0) to the monoid (N , ×, 1), defined by f (x) = 2x . It is injective, but not surjective.
7.2
Informal discussion
Because abstract algebra studies sets endowed with operations that generate interesting structure or properThe real numbers are a ring, having both addition and ties on the set, functions which preserve the operations multiplication. The set of all 2 × 2 matrices is also a ring, are especially important. These functions are known as under matrix addition and matrix multiplication. If we homomorphisms. define a function between these rings as follows: For example, consider the natural numbers with addition as the operation. A function which preserves addition should have this property: f (a + b ) = f (a) + f (b). For r 0 example, f ( x ) = 3 x is one such homomorphism, since f (a f (r ) = 0 r + b) = 3(a + b) = 3a + 3b = f (a) + f (b). Note that this homomorphism maps the natural numbers back into themwhere r is a real number, then f is a homomorphism of selves. rings, since f preserves both addition: Homomorphisms do not have to map between sets which have the same operations. For example, operationpreserving functions exist between the set of real numr + s r 0 s 0 0 f (r +s) = f (s) with addition and the positive real numbers ℝ + = + = f (r )+ bers ℝ r + s 0 0 r 0 s with multiplication. A function which preserves operation should have this property: f (a + b ) = f (a) · f (b), and multiplication: since addition is the operation in the first set and multiplication is the operation in the second. Given the laws of exponents, f ( x ) = e x satisfies this condition: 2 + 3 = 5 rs 0 r 0 s 0 f (rs ) = = = f (r ) f (s). translates into e2 · e3 = e5 . 0 rs 0 r 0 s If we are considering multiple operations on a set, then For another example, the nonzero complex numbers form all operations must be preserved for a function to be cona group under the operation of multiplication, as do the sidered as a homomorphism. Even though the set may be nonzero real numbers. (Zero must be excluded from both the same, the same function might be a group homomorgroups since it does not have a multiplicative inverse, phism, (a single binary operation, an inverse operation, which is required for elements of a group.) Define a func- being a unary operation, and identity, being a nullary option f from the nonzero complex numbers to the nonzero eration) but not a ring isomorphism (two binary operareal numbers by tions, the additive inverse and the identity elements), because it may fail to preserve the additional monoid struc f ( z) = |z|. ture required by the definition of a ring.
� � �
� � �� �
� � � �� �
50
CHAPTER 7. HOMOMORPHISM
Hom
Mon
Iso
Epi
and only if it is both injective and surjective, in abstract algebra a homomorphism is an isomorphism if and only if it is both a monomorphism and an epimorphism. An isomorphism always has an inverse f −1 , which is a homomorphism, too (cf. Proof 1). If there is an isomorphism between two algebraic structures, they are completely indistinguishable as far as the structure in question is concerned; in this case, they are said to be isomorphic .
Aut ∞
∞
End
Relationships between different kinds of homomorphisms. Hom = set of Homomorphisms, Mon = set of Monomorphisms, Epi = set of Epimorphisms, Iso = set of Isomorphisms, End = set of Endomorphism, Aut = set of Automorphisms. Notice that: Mon ∩ Epi = Iso , Iso ∩ End = Aut. The sets (Mon ∩ End) \ Aut and (Epi ∩ End) \ Aut contain only homomorphisms from some infinite structures to themselves.
7.3 Types In abstract algebra, several specific kinds of homomorphisms are defined as follows:
• An isomorphism is a bijective homomorphism. • An epimorphism (sometimes called a cover) is a [note 1]
7.3.1
Category theory
Since homomorphisms are morphisms in an appropriate category, we mayconsider the analogous specific kinds of morphisms defined in any category. However, the definitions in category theory are somewhat different. For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions; the first three descriptions do not. In categorytheory, a morphism f : A → B is called:
• monomorphism if f ∘ g1 = f ∘ g2 implies g1 = g2
for all morphisms g 1, g2 : X → A, where "∘" denotes function composition corresponding to e.g. ( f ∘ g1 )( x ) = f ( g1 ( x )) in abstract algebra. (A sufficient condition for this is f having a left inverse, cf. Proof 2.)
• epimorphism if g1 ∘ f = g2 ∘ f implies g1 = g2 for all morphisms g1 , g2 : B → X . (A sufficient condition for this is f having a right inverse, cf. Proof 3.)
• isomorphism if there exists a morphism g: B → A
such that f ∘ g = 1B and g ∘ f = 1A, where “1X " denotes theidentitymorphism on theobject X .[note 2]
For instance, the inclusion ring homomorphism of Z as a (unitary) subring of Q is not surjective (i.e. not epi in surjective homomorphism. Equivalently, f : A the set-theoretic sense), but an epimorphic in the sense → B is an epimorphism if it has a right inverse g: B of category theory.[3][4] This inclusion thus also is an ex→ A, i.e. if f ( g(b)) = b for all b ∈ B. ample of a ring homomorphism which is (in the sense of • A monomorphism (sometimes called an category theory) both mono and epi, but not iso. embedding or extension) is an injective homomorphism. Equivalently, [note 1] f : A → B is a monomorphism if it has a left inverse g: B → A, i.e. 7.4 Kernel if g( f (a)) = a for all a ∈ A. • An endomorphism is a homomorphism from an al- Main article: Kernel (algebra) gebraic structure to itself. homomorphism f : X → Y defines an equivalence • An automorphism is an endomorphism which is Any relation ~ on X by a ~ b if and only if f (a) = f (b). The also an isomorphism, i.e., an isomorphism from an relation ~ is called the kernel of f . It is a congruence algebraic structure to itself. [1] relation on X . The quotient set X / ~ can then be given an • The trivial homomorphism between unital mag- object-structure in a natural way, i.e. [ x ] ∗ [y] = [ x ∗ y]. In mas is the constant map onto the identity element of that case the image of X in Y under the homomorphism the codomain.[2] f is necessarily isomorphic to X / ~; this fact is one of the isomorphism theorems. Note in some cases (e.g. groups These descriptions may be used in order to derive several or rings), a single equivalence class K suffices to specify properties. For instance, since a function is bijective if the structure of the quotient, in which case we can write
51
7.8. NOTES
it X /K . (X /K is usually read as " X mod K ".) Also in these 7.8 Notes cases, it is K , rather than ~, that is called the kernel of f (cf. normal subgroup). [1] tacitly assuming the axiom of choice and a nonconstructive setting
7.5
Relational structures
[2] The notion of “object” and“morphism” in categorytheory generalizes the notion of “algebraic structure” and “homomorphism”, respectively.
In model theory, the notion of an algebraic structure is [3] In homomorphisms on formal languages, the ∗ operageneralized to structures involving both operations and tion is the Kleene star operation. The ⋅ and ∘ are both relations. Let L be a signature consisting of function and concatenation, commonly denoted by juxtaposition. relation symbols, and A , B be two L -structures. Then a homomorphism from A to B is a mapping h from the domain of A to the domain of B such that 7.9 References
• h(F A(a1,…,an)) = F B (h(a1),…,h(an)) for each n ary function symbol F in L,
• RA (a1,…,an) implies RB(h(a1),…,h(an)) for each n-ary relation symbol R in L.
[1] Birkhoff, Garrett (1967)[1940], Lattice theory, American Mathematical Society Colloquium Publications 25 (3rd ed.), Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1025-5, MR 598630 Here: Sect.VI.3, p.134 [2] Bourbaki, Algebra, ch. I §2.1, p. 13
In thespecial case with just one binary relation, we obtain the notion of a graph homomorphism. For a detailed discussion of relational homomorphisms and isomorphisms see.[5]
[3] Mac Lane, Saunders (1971). Categories for the Working Mathematician. Graduate Texts in Mathematics 5. Springer-Verlag. Exercise 4 in section I.5. ISBN 0-38790036-5. Zbl 0232.18001.
7.6
[4] Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Şerban (2001). Hopf Algebra: An Introduction. Pure and Applied Mathematics 235. New York, NY: Marcel Dekker. p. 363. ISBN 0824704819. Zbl 0962.16026.
Formal language theory
Homomorphisms are also used in the study of formal [5] Section 17.4, inGunther Schmidt, 2010. Relational Mathlanguages[6] (although within this context, often they are ematics. Cambridge University Press, ISBN 978-0-521briefly referred to as morphisms[7] ). Given alphabets Σ1 76268-7 ∗ ∗ and Σ2 , a function h : Σ1 → Σ2 such that h(uv) = h(u) h(v) for all u and v in Σ1 ∗ is called a homomorphism (or [6] Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, simply morphism) on Σ1 ∗ .[note 3] Let e denote the empty ISBN 0-7204-2506-9. word. If h is a homomorphism on Σ1 ∗ and h( x ) ≠ e for all x ≠ e in Σ1 ∗ , then h is called an e-free homomorphism . [7] T. Harju, J. Karhumӓki, Morphisms in Handbook of Formal Languages, Volume I, edited by G. Rozenberg, A. SaThis type of homomorphism can be thought of as (and is lomaa, Springer, 1997, ISBN 3-540-61486-9. equivalent to) a monoid homomorphism where Σ∗ the set of all words over a finite alphabet Σ is a monoid (in fact it is the free monoid on Σ) with operation concatenation A monograph available free online: and the empty word as the identity. • Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. SpringerVerlag. ISBN 3-540-90578-2.
7.7
See also
• Continuous function • Diffeomorphism • Homomorphic encryption • Homomorphic secret sharing – a simplistic decentralized voting protocol
• Morphism
Chapter 8
Ideal In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any other integer results in another even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring similarly to the way that, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may be distinct from the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory). The concept of an order ideal in order theory is derived from the notion of ideal in ring theory. A fractional ideal is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity.
ply an ideal) of R if it is an additive subgroup of R that “absorbs multiplication by elements of R". Formally we mean that I is an ideal if it satisfies the following conditions: 1. (I, +) is a subgroup of (R, +) 2.
∀x ∈ I, ∀r ∈ R :
·
· ∈ I
x r, r x
Equivalently, an ideal of R is a sub-R-bimodule of R. A subset I of R is called a right ideal of R [3] if it is an additive subgroup of R and absorbs multiplication on the right, that is: 1. (I, +) is a subgroup of (R, +) 2.
∀x ∈ I, ∀r ∈ R :
· ∈ I.
x r
Equivalently, a right ideal of R is a right R -submodule of R . Similarly a subset I of R is called a left ideal of R if it is an additive subgroup of R absorbing multiplication on the left: 1. (I, +) is a subgroup of (R, +) 2.
∀x ∈ I, ∀r ∈ R :
· ∈ I.
r x
Equivalently, a left ideal of R is a left R -submodule of R . 8.1 History In all cases, the first condition can be replaced by the folIdeals were first proposed by Richard Dedekind in 1876 lowing well-known criterion that ensuresa nonemptysubin the third edition of his book Vorlesungen über Zahlen- set of a group is a subgroup: theorie (English: Lectures on NumberTheory). Theywere 1'. I is non-empty and ∀x, y ∈ I : x − y ∈ I a generalization of the concept of ideal numbers devel.[4] oped by Ernst Kummer.[1][2] Later the concept was expanded by David Hilbert and especially Emmy Noether. The left ideals in R are exactly the right ideals in the opposite ring R o and vice versa. A two-sided ideal is a left ideal that is also a right ideal, and is often called an 8.2 Definitions ideal except to emphasize that there might exist singlesided ideals. When R is a commutative ring, the definiFor an arbitrary ring (R, +, ·) , let (R, +) be its additive tions of left, right, and two-sided ideal coincide, and the group. A subset I is called a two-sided ideal (or sim- term ideal is used alone. 52
53
8.6. IDEAL GENERATED BY A SET
8.3 Properties {0} and R are ideals in every ring R . If R is a division ring or a field, then these are its only ideals. The ideal R is called the unit ideal. I is a proper ideal ifitisaproper subset of R, that is, I does not equal R.[5] Just as normal subgroups of groups are kernels of group homomorphisms, ideals have interpretations as kernels. For a nonempty subset A of R:
• A is an ideal of R if and only if it is a kernel of a ring
forms an ideal. These two ideals are usually referred to as the trivial ideals of R.
• The even integers form an ideal in the ring Z of all
integers; it is usually denoted by 2Z . This is because the sum of any even integers is even, and the product of any integer with an even integer is also even. Similarly, the set of all integers divisible by a fixed integer n is an ideal denoted nZ .
• The set of all polynomials with real coefficients 2
which are divisible by the polynomial x + 1 is an ideal in the ring of all polynomials.
homomorphism from R.
• A is a right ideal of R if and only if it is a kernel
of a homomorphism from the right R module RR to another right R module.
• The set of all n-by-n matrices whose last row is zero
forms a right ideal in the ring of all n-by-n matrices. It is not a left ideal. The set of all n -by-n matrices whose last column is zero forms a left ideal but not a right ideal.
• A is a left ideal of R if and only if it is a kernel of
a homomorphism from the left R module RR to another left R module.
If p is in R, then pR is a right ideal and Rp is a left ideal of R. These are called, respectively, the principal right and left ideals generated by p. To remember which is which, note that right ideals are stable under right-multiplication (IR ⊆ I ) and left ideals are stable under left-multiplication (RI ⊆ I ). The connection between cosets and ideals can be seen by switching the operation from “multiplication” to “addition”.
8.4 Motivation Intuitively, the definition can be motivated as follows: Suppose we have a subset of elements Z of a ring R and that wewould like to obtain a ring with the same structure as R , except that the elements of Z should be zero (they are in some sense “negligible”). But if z 1 = 0 and z 2 = 0 in our new ring, then surely z1 + z2 should be zero too, and rz 1 as well as z1 r should be zero for any element r (zero or not). The definition of an ideal is such that the ideal I generated (see below) by Z is exactly the set of elements that are forced to become zero if Z becomes zero, and the quotient ring R/I is the desired ring where Z is zero, and only elements that areforcedby Z to bezeroare zero. The requirement that R and R/I shouldhave the samestructure (except that I becomes zero) is formalized by the condition that the projection from R to R/I is a (surjective) ring homomorphism.
8.5
• The ring C (R) of all continuous functions f from R to R under pointwise
multiplication contains the ideal of all continuous functions f such that f (1) = 0. Another ideal in C (R) is given by those functions which vanish for large enough arguments, i.e. those continuous functions f for which there exists a number L > 0 such that f ( x ) = 0 whenever |x | > L.
• Compact operators form an ideal in the ring of bounded operators.
8.6
Ideal generated by a set
Let R be a (possibly not unital) ring. Any intersection of any nonempty family of left ideals of R is again a left ideal of R . If X is any subset of R , then the intersection of all left ideals of R containing X is a left ideal I of R containing X , and is clearly the smallest left ideal to do so. This ideal I is said to be the left ideal generated by X . Similar definitions can be created by using right ideals or two-sided ideals in place of left ideals. If R has unity, then the left, right, or two-sided ideal of R generated by a subset X of R can be expressed internally as we will now describe. The following set is a left ideal:
{r1x1 + ··· + r x | n ∈ N, r ∈ R, x ∈ X }. n n
i
i
Each element described would have to be in every left ideal containing X , so this left ideal is in fact the left ideal generated by X . The right ideal and ideal generated by X can also be expressed in the same way:
Examples
• In a ring R, the set R itself forms an ideal of R. Also,
the subset containing only the additive identity 0 R
{x1r1 + ··· + x r | n ∈ N, r ∈ R, x ∈ X } {r1x1s1+···+r x s | n ∈ N, r ∈ R, s ∈ R, x ∈ X }. n n
n n n
i
i
i
i
i
54
CHAPTER 8. IDEAL
Theformer is therightideal generated by X , andthelatter is the ideal generated by X . By convention, 0 is viewed as the sum of zero such terms, agreeing with the fact that the ideal of R generated by ∅ is {0} by the previous definition. If a left ideal I of R has a finite subset F such that I is the left ideal generated by F , then the left ideal I is said to be finitely generated. Similar terms are also applied to right ideals and two-sided ideals generated by finite subsets. In the special case where the set X is just a singleton {a} for some a in R , then the above definitions turn into the following:
• Maximal ideal: A proper ideal I is called a maxi-
mal ideal if there exists no other proper ideal J with
I a proper subset of J . The factor ring of a maximal
idealisa simple ring ingeneralandisa field for commutative rings.[6]
• Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
• Prime ideal:
A proper ideal I is called a prime ideal if for any a and b in R, if ab is in I , then at least oneof a and b isin I . The factor ring of a prime ideal is a prime ring in general and is an integral domain for commutative rings.
• Radical ideal or semiprime ideal: A proper ideal I is called radical or semiprime if for any a in R, if an is in I for some n, then a is in I . The factor ring of a radical ideal is a semiprime ring for general rings, and is a reduced ring for commutative rings.
Ra = ra r
{ | ∈ R} aR = {ar | r ∈ R} RaR = {r1 as1 +···+r
| n ∈ N, r ∈ R, s ∈ R}.
n asn
i
i
These ideals are known as the left/right/two-sided principal ideals generated by a . It is also very common to denote the two-sided ideal generated by a as (a). If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form x + x +...+ x , and n-foldsumsoftheform(−x )+(−x )+...+(−x )forevery x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous. 8.6.1
Example
• In the ring Z of integers, every ideal can be gener-
ated by a single number (so Z is a principal ideal domain), and the only two generators of pR are p and −p. The concepts of “ideal” and “number” are therefore almost identical in Z . If aR = bR in an arbitrary domain, then au = b for some unit u. Conversely, for any unit u, aR = auu−1 R = auR. So, in a commutative principal ideal domain, the generators of the ideal aR are just the elements au where u is an arbitrary unit. This explains the case of Z since 1 and −1 are the only units of Z .
• Primary ideal : An ideal I is called a primary ideal
if for all a and b in R, if ab is in I , then at least one of a and b n is in I for some natural number n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.
• Principal ideal: An ideal generated by one element. • Finitely generated ideal: This type of ideal is finitely generated as a module.
A left primitive ideal is the annihilator of a simple left module. A right primitive ideal is defined similarly. Actually (despite the name) the left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. A factor rings constructed with a right (left) primitive ideals is a right (left) primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields.
• Primitive
ideal:
• Irreducible ideal : An ideal is said to be irreducible if it cannot be written as an intersection of ideals which properly contain it.
• Comaximal ideals: Two ideals i, j are said to be comaximal if x + y = 1 for some x ∈ i and y ∈ j . • Regular ideal: This term hasmultiple uses. See the article for a list.
8.7 Types of ideals
• Nil ideal: An ideal is a nil ideal if each of its elements is nilpotent.
To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles.
Ideals are important because they appear as kernels of ring homomorphisms and allow one to define factor rings. Different types of ideals are studied because they can be used to construct different types of factor rings.
Two other important terms using “ideal” are not always ideals of their ring. See their respective articles for details:
• Fractional ideal: This is usuallydefined when R is a commutative domain with quotient field K . Despite their names, fractional ideals are R submodules of
55
8.10. IDEALS AND CONGRUENCE RELATIONS K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.
• Invertible ideal :
| ∈ a and b ∈ b}
a + b := a + b a
{
Usually an invertible ideal A is defined as a fractional ideal for which there is an- and other fractional ideal B such that AB=BA=R. Some authors may also apply “invertible ideal” to ordinary ring ideals A and B with AB =BA=R in rings other ab := {a1 b1 +···+an bn | ai ∈ a and bi ∈ b, i = 1, 2, . . . , n; for n = 1, 2 than domains. i.e. the product of two ideals a and b is defined to be the ideal ab generated by all products of the form ab with a in a and b in b . The product ab is contained in the 8.8 Further properties intersection of a and b . • In rings with identity, an ideal is proper if and only The sum and the intersection of ideals is again an ideal; if it does not contain 1 or equivalently it does not with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular latcontain a unit. tice. Also, the union of two ideals is a subset of the sum • The set of ideals of any ring are partially ordered of those two ideals, because for any element a inside an via subset inclusion, in fact they are additionally a ideal, we can write it as a+0, or 0+a, therefore, it is concomplete modular lattice in this order with join op- tained in the sum as well. However, the union of two eration given by addition of ideals and meet opera- ideals is not necessarily an ideal. tion given by set intersection. The trivial ideals supply the least and greatest elements: the largest ideal is the entire ring, and the smallest ideal is the zero 8.10 Ideals and congruence relaideal. The lattice is not, in general, a distributive tions lattice.
• Unfortunately Zorn’s lemma does not necessarily
apply to the collection of proper ideals of R . However, when R has identity 1, this collection can be reexpressed as “the collection of ideals which do not contain 1”. It can be checked that Zorn’slemmanow applies to this collection, and consequently there are maximal proper ideals of R. With a little more work, it can be shown that every proper ideal is contained in a maximal ideal. See Krull’s theorem at maximal ideal.
• The ring R can be considered as a left module over
itself, and the left ideals of R are then seen as the submodules of this module. Similarly, the right ideals are submodules of R as a right module over itself, and the two-sided ideals are submodules of R as a bimodule over itself. If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.
• Every ideal is a pseudo-ring. • The ideals of a ring form a semiring (with identity element R) under addition and multiplication of ideals.
8.9
Ideal operations
The sum and product of ideals are defined as follows. For a and b , ideals of a ring R,
There is a bijective correspondence between ideals and congruence relations (equivalence relations that respect the ring structure) on the ring: Given an ideal I of a ring R, let x ~ y if x − y ∈ I . Then ~ is a congruence relation on R. Conversely, given a congruence relation ~ on R, let I = { x : x ~ 0}. Then I is an ideal of R.
8.11 See also
• Modular arithmetic • Noether isomorphism theorem • Boolean prime ideal theorem • Ideal theory • Ideal (order theory) • Ideal quotient • Ideal norm • Artinian ideal • Noncommutative ring • Regular ideal • Idealizer
56
CHAPTER 8. IDEAL
8.12 References [1] Harold M. Edwards (1977). Fermat’s last theorem. A genetic introduction to algebraic number theory . p. 76. [2] Everest G., Ward T. (2005). An introduction to number theory. p. 83. [3] See Hazewinkel et al. (2004), p. 4. [4] In fact, since R is assumed to be unital, it suffices that x + y is in I , since the second condition implies that −y is in I . [5] Lang 2005, Section III.2 [6] Because simple commutative rings are fields. See Lam (2001). A First Course in Noncommutative Rings. p. 39.
• Lang, Serge (2005). Undergraduate Algebra (Third ed.). Springer-Verlag. ISBN 978-0-387-22025-3
• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda
Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules . Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0
Chapter 9
Integral domain • An integral domain is a commutative ring in which In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the zero ideal {0} is a prime ideal. the product of any two nonzero elements is nonzero. [1][2] • An integral domain is a nonzero commutative ring Integral domains are generalizations of the ring of intefor which every non-zero element is cancellable ungers and provide a natural setting for studying divisibility. der multiplication. In an integral domain the cancellation property holds for multiplication by a nonzero element a, that is, if a ≠ 0, an • An integral domain is a ring for which the set of equality ab = ac implies b = c . nonzero elements is a commutative monoid under multiplication (because the monoid is closed under “Integral domain” is defined almost universally as above, multiplication). but there is some variation. This article follows the convention that rings have a multiplicative identity, gen• An integral domain is a ring that is (isomorphic to) erally denoted 1, but some authors do not follow this, a subring of a field. (This implies it is a nonzero by not requiring integral domains to have a multiplicacommutative ring.) tive identity.[3][4] Noncommutative integral domains are sometimes admitted.[5] This article, however, follows • An integral domain is a nonzero commutative ring the much more usual convention of reserving the term in which for every nonzero element r , the function “integral domain” for the commutative case and using that maps each element x of the ring to the prod"domain" for the general case including noncommutative uct xr is injective. Elements r with this property are rings. called regular , so it isequivalent to require that every nonzero element of the ring be regular. Some sources, notably Lang, use the term entire ring for [6] integral domain. Some specific kinds of integral domains are given with 9.2 Examples the following chain of class inclusions: • The
archetypical example is the ring Z of all integers.
commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields
• Every field is an integral domain. Conversely, every
Artinian integral domain is a field. In particular, all finite integral domains are finite fields (more generally, by Wedderburn’s little theorem, finite domains are finite fields). The ring of integers Z provides an example of a non-Artinian infinite integral domain that is not a field, possessing infinite descending sequences of ideals such as:
9.1 Definitions There are a number of equivalent definitions of integral domain:
Z
• An integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
⊃
2Z
⊃ ··· ⊃
2n Z
⊃
2n+1 Z
⊃ ···
• Rings of polynomials are integral domains if the co-
efficients come from an integral domain. For instance, the ring Z[X ] of all polynomials in one variable with integer coefficients is an integral domain;
• An integral domain is a nonzero commutative ring with no nonzero zero divisors.
57
58
CHAPTER 9. INTEGRAL DOMAIN
so is the ring R[X ,Y ] of all polynomials in two vari- If a divides b and b divides a, then we say a and b are asables with real coefficients. sociated elements or associates.[9] Equivalently, a and • For each integer n > 1, the set of all real numbers of b are associates if a=ub for some unit u. the form a + b √n with a and b integers is a subring If q is a nonzero non-unit, we say that q is an irreducible element if q cannot be written as a product of two nonof R and hence an integral domain. units. • For each integer n >0thesetofall complex numbers If p is a nonzero non-unit, we say that p is a prime eleoftheform a + bi √n with a and b integersis a subring ment if, whenever p divides a product ab, then p divides a of C and hence an integral domain. In the case n = 1 or p divides b. Equivalently, an element p is prime if and this integral domain is called the Gaussian integers. only if the principal ideal ( p) is a nonzero prime ideal. • The ring of p-adic integers is an integral domain. The notion of prime element generalizes the ordinary definition of prime number in the ring Z , except that it • If U is a connected opensubset ofthe complex plane allows for negative prime elements. C, then the ring H(U ) consisting of all holomorphic functions f : U → C is an integral domain. The same Every prime element is irreducible. The converse is not in general: for example, in the quadratic integer ring is true for rings of analytic functions on connected true√ −5 the element 3 is irreducible (if it factored nonZ open subsets of analytic manifolds. trivially, the factors would each have to have norm 3, • A regular local ring is an integral domain. In fact, a but there are no norm 3 elements since a2 + 5b2 = 3 has no√ integer solutions), but not prime (since 3 divides regular local ring is a UFD.[7][8] √ 2 + −5 2 − −5 without dividing either factor). In a unique factorization domain (or more generally, a GCD domain), an irreducible element is a prime element. 9.3 Non-examples √ While unique factorization does not hold in Z −5 , there is unique factorization of ideals. See Lasker– The following rings are not integral domains. Noether theorem.
� �
� �
• The ring of n × n matrices over any nonzero ring when n ≥ 2.
• Theringof continuous functions onthe unit interval. • The quotient ring Z/mZ when m is a composite number.
• The product ring Z × Z. • The zero ring in which 0=1. • The tensor product C ⊗ C (since, for example, (i ⊗ 1 − 1 ⊗ i) (i ⊗ 1 + 1 ⊗ i) = 0 ). • The quotient ring k[x, y]/(xy) for any field k , since
9.5 Properties
• A commutative ring R is an integral domain if and only if the ideal (0) of R is a prime ideal.
• If R is a commutative ring and P is an ideal in R, then the quotient ring R/P is an integral domain if and only if P is a prime ideal.
R
(xy ) is not a prime ideal.
9.4
Divisibility, prime elements, and irreducible elements
• Let R be an integral domain. Then there is an integral domain S such that R ⊂ S and S has an element which is transcendental over R.
• The cancellation property holds in any integral do-
main: for any a , b , and c in an integral domain, if a ≠ 0 and ab = ac then b = c . Another way to state this is that the function x ↦ ax is injective for any nonzero a in the domain.
See also: Divisibility (ring theory) In this section, R is an integral domain. Given elements a and b of R, we say that a divides b, or that a is a divisor of b , or that b is a multiple of a , if there exists an element x in R such that ax = b. The elements that divide 1 are called the units of R; these are precisely the invertible elements in R. Units divide all other elements.
• The cancellation property holds for ideals in any in-
tegral domain: if xI = xJ , then either x is zero or I = J .
• An integral domain is equal to the intersection of its localizations at maximal ideals.
• An inductive limit of integral domains is an integral domain.
59
9.10. NOTES
9.6
Field of fractions
Main article: Field of fractions
9.10
Notes
[1] Bourbaki, p. 116. [2] Dummit and Foote, p. 228.
The field of fractions K of an integral domain R is the set of fractions a /b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. It is “the smallest field containing R " in the sense that there is an injective ring homomorphism R → K such that any injective ring homomorphism from R to a field factors through K . The field of fractions of the ring of integers Z is the field of rational numbers Q. The field of fractions of a field is isomorphic to the field itself.
9.7
Algebraic geometry
[3] B.L. van der Waerden, Algebra Erster Teil, p. 36, Springer-Verlag, Berlin, Heidelberg 1966. [4] I.N. Herstein, Topics in Algebra, p. 88-90, Blaisdell Publishing Company, London 1964. [5] J.C. McConnel and J.C. Robson “Noncommutative NoetherianRings” (Graduate Studies in MathematicsVol. 30, AMS) [6] Pages 91–92 of Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley Pub. Co., ISBN 978-0201-55540-0, Zbl 0848.13001 [7] Auslander, Maurice; Buchsbaum, D. A. (1959). “Unique factorization in regular local rings”. Proc. Natl. Acad. Sci. USA 45 (5): 733–734. doi:10.1073/pnas.45.5.733. PMC 222624. PMID 16590434.
Integral domains are characterized by the condition that they are reduced (that is x 2 = 0 implies x = 0) and irreducible (that is there is only one minimal prime ideal). [8] Masayoshi Nagata (1958). “A general theory of algebraic geometry over Dedekind domains. II”. Amer. J. Math. The former condition ensures that the nilradical of the (The Johns Hopkins University Press) 80 (2): 382–420. ring is zero, so that the intersection of all the ring’s mindoi:10.2307/2372791. JSTOR 2372791. imal primes is zero. The latter condition is that the ring have only one minimal prime. It follows that the unique [9] Durbin, John R. (1993). Modern Algebra: An Introduction minimal prime ideal of a reduced and irreducible ring is (3rd ed.). John Wiley and Sons. p. 224. ISBN 0-471the zero ideal, so such rings are integral domains. The 51001-7. Elements a and b of [an integral domain] are converse is clear: an integral domain has no nonzero called associates if a | b and b | a. nilpotent elements, and the zero ideal is the unique minimal prime ideal. This translates, in algebraic geometry, into the fact that 9.11 References the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is an algebraic • Adamson, Iain T. (1972). Elementary rings and variety. modules. University Mathematical Texts. Oliver and Boyd. ISBN 0-05-002192-3. More generally, a commutative ring is an integral domain if and only if its spectrum is an integral affine scheme. • Bourbaki, Nicolas (1998). Algebra, Chapters 1–3 . Berlin, New York: Springer-Verlag. ISBN 978-3540-64243-5.
9.8 Characteristic and homomorphisms
• Mac Lane, Saunders; Birkhoff, Garrett (1967). Al gebra. New York: The Macmillan Co. ISBN 1-
56881-068-7. MR 0214415.
The characteristic of an integral domain is either 0 or a prime number. If R is an integral domain of prime characteristic p, then the Frobenius endomorphism f ( x ) = x p is injective.
9.9
See also
• Dedekind–Hasse norm – the extra structure needed for an integral domain to be principal
• Zero-product property
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). New York: Wiley. ISBN
978-0-471-43334-7.
• Hungerford, Thomas W. (1974). Algebra.
New York: Holt, Rinehart and Winston, Inc. ISBN 003-030558-6.
• Lang, Serge (2002). Algebra.
Graduate Texts in Mathematics 211. Berlin, New York: SpringerVerlag. ISBN 978-0-387-95385-4. MR 1878556.
• Sharpe, David (1987).
Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.
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CHAPTER 9. INTEGRAL DOMAIN
• Rowen, Louis Halle (1994). Algebra: groups, rings, and fields. A K Peters. ISBN 1-56881-028-8.
• Lanski, Charles (2005). Concepts in abstract algebra. AMS Bookstore. ISBN 0-534-42323-X.
• Milies, César Polcino; Sehgal, Sudarshan K. (2002).
An introduction to group rings . Springer. ISBN 1-
4020-0238-6.
• B.L. van der Waerden, Algebra, Springer-Verlag, Berlin Heidelberg, 1966.
Chapter 10
Isometry This article is about distance-preserving functions. For a composition of a rigid motion and a reflection. other mathematical uses, see isometry (disambiguation). Isometries are often used in constructions where one For non-mathematical uses, see Isometric. space is embedded in another space. For instance, the completion of a metric space M involves an isometry In mathematics, an isometry (or congruence, or from M into M' , a quotient set of the space of Cauchy secongruent transformation) is a distance-preserving quences on M . The original space M is thus isometrically injective map between metric spaces.[1] isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is iso A R 1 ( A ) R2 ( A 1 ) A1 A2 metrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isoD metrically isomorphic to a closed subset of some Banach S space. C B An isometric surjective linear operator on a Hilbert space is called a unitary operator. A 2
T C2
R1
T
B1
Let X and Y be metric spaces with metrics dX and dY . A map ƒ : X → Y is called an isometry or distance preserving if for any a,b ∈ X one has
A 1
Δ 1 R 2
10.2 Formal definitions
B2
Δ 2
C1
D1
S
S
dY (f (a), f (b)) = d X (a, b). [4]
T
A composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like R1 or R2 on the image. Translation T is a direct isometry: a rigid motion.[2]
10.1
Introduction
Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they arerelated byan isometry;[3] the isometrythat relates them is either a rigid motion (translation or rotation), or
An isometry is automatically injective;[1] otherwise two distinct points, a and b , could be mapped to the same point, thereby contradicting the coincidence axiom of the metric d . This proof is similar to the proof that an order embedding between partially ordered sets is injective. Clearly, every isometry between metric spaces is a topological embedding (i.e. a homeomorphism). A global isometry , isometric isomorphism or congruence mapping is a bijective isometry. Like any other bijection, a global isometry has a function inverse. The inverse of a global isometry is also a global isometry. Two metric spaces X and Y are called isometric if there is a bijective isometry from X to Y . The set of bijective isometries from a metric space to itself forms a group with respect to function composition, called the isometry group. There is also the weaker notion of path isometry or arcwise isometry:
61
62
CHAPTER 10. ISOMETRY
A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
further than ε away from the image of an element of the domain. Note that ε-isometries are not assumed to be continuous.
• The restricted isometry property characterizes nearly isometric matrices for sparse vectors.
• Quasi-isometry is yet another useful generalization.
10.3
Examples
• Any reflection, translation and rotation is a global
isometry on Euclidean spaces. See also Euclidean group.
• The map x → | x| in R is a path isometry but not an isometry. Note that unlike an isometry, it is not injective.
• The isometric linear maps[5][6][7][8] from Cn to itself aregiven by the unitary matrices.
10.4 Linear isometry
• One may also define an element in an abstract unital C*-algebra to be an isometry:
a A is an isometry if and only if a ∗ a = 1 .
∈
·
Note that as mentioned in the introduction this is notnecessarily a unitary element because one does not in general have that left inverse is a right inverse.
• On a pseudo-Euclidean space, the term isometry means a linear bijection preserving magnitude. See also Quadratic spaces.
Given two normed vector spaces V and W , a linear isom- 10.6 See also etry is a linear map f : V → W that preserves the norms: • Motion (geometry)
∥f (v)∥ = ∥v∥ for all v in V . Linear isometries are distance-preserving maps in the above sense. They are global isometries if and only if they are surjective. By the Mazur-Ulam theorem, any isometry of normed vector spaces over R is affine. In an inner product space, the fact that any linear isometry is an orthogonal transformation canbe shown by using polarization to prove
= and then applying the Riesz representation theorem.
10.5 Generalizations
• Given a positive real number ε, an ε-isometry or almost isometry (also called a Hausdorff approximation) is a map f : X → Y between metric
• Beckman–Quarles theorem • Semidefinite embedding • Flat (geometry) • Euclidean plane isometry • 3D isometries that leave the origin fixed • Space group • Involution • Symmetry in mathematics • Homeomorphism group • Partial isometry • The second dual of a Banach space as an isometric isomorphism
spaces such that
1. for x , x ′ ∈ X one has |dY (ƒ( x ),ƒ( x ′))−dX ( x , x ′)| 10.7 References < ε, and [1] Coxeter 1969, p. 29 2. for any point y ∈ Y there exists a point x ∈ X “We shall find it convenient to usetheword transformation with dY (y,ƒ( x )) < ε That is, an ε-isometry preserves distances to within ε andleaves no element of thecodomain
in the special sense of a one-to-one correspondence P → P ′ among all points in the plane (or in space), that is, a rule for associating pairs of points, withthe understanding that each pair has a first member P and a second member
10.8. BIBLIOGRAPHY P' and that every point occurs as the first member of just one pair and also as the second member of just one pair... In particular, an isometry (or “congruent transformation,” or “congruence”) is a transformation which preserves length...” [2] Coxeter 1969, p. 46 3.51 Any direct isometry is either a translation or a rota-
tion. Any opposite isometry is either a reflection or a glide reflection.
[3] Coxeter 1969, p. 39 3.11 Any two congruent triangles are related by a unique
isometry.
[4] Beckman, F. S.; Quarles, D. A., Jr. (1953). “On isometries of Euclidean spaces” (PDF). Proceedings of the American Mathematical Society 4: 810–815. doi:10.2307/2032415. MR 0058193. Let T be a transformation (possibly many-valued) of E n ( 2 ≤ n < ∞ ) into itself. Let d ( p, q ) be the distance between points p and q of E n , and let Tp, Tq be any images of p and q, respectively. If there is a length a > 0 such that d (T p , T q) = a whenever d ( p, q ) = a , then T is a Euclidean transformation of E n onto itself. [5] Roweis, S. T.; Saul, L. K. (2000). “Nonlinear Dimensionality Reduction by Locally Linear Embedding”. Science 290 (5500): 2323–2326. doi:10.1126/science.290.5500.2323. PMID 11125150. [6] Saul, Lawrence K.; Roweis, Sam T. (2003). “Think globally, fit locally: Unsupervised learning of nonlinear manifolds". Journal of Machine Learning Research (http: //jmlr.org/papers/v4/saul03a.html) 4 (June): 119–155. Quadraticoptimisation of M = (I − W )⊤ (I − W ) (page 135) such that M ≡ Y Y ⊤ [7] Zhang, Zhenyue; Zha, Hongyuan (2004). “Principal Manifolds and Nonlinear Dimension Reduction via Local Tangent Space Alignment". SIAM Journal on Scientific Computing 26 (1): 313–338. doi:10.1137/s1064827502419154. [8] Zhang, Zhenyue; Wang, Jing (2006). “MLLE: Modified Locally Linear Embedding Using Multiple Weights”. Advances in Neural Information Processing Systems 19 . It can retrieve the ideal embedding if MLLE is applied on data points sampled from an isometric manifold.
10.8 Bibliography
• Coxeter, H. S. M. (1969). Introduction to Geometry, Second edition. Wiley. ISBN 9780471504580.
63
Chapter 11
Magma For other uses, see Magma (disambiguation).
b. The symbol, •, is a general placeholder for a properly
defined operation. To qualify as a magma, the set and In abstract algebra, a magma (or groupoid; not to be operation (M , •) must satisfy the following requirement confused with groupoids in category theory) is a basic (known as the magma or closure axiom ): kind of algebraic structure. Specifically, a magma conFor all a, b in M , the result of the operation a • sists of a set, M , equipped with a single binary operation, b is also in M . M × M → M . The binary operation must be closed by definition but no other properties are imposed. And in mathematical notation: ∀ a, b ∈ M : a • b ∈ M .
11.1 History and terminology The term groupoid was introduced in 1926 by Heinrich Brandt describing his Brandt groupoid (translated from the German Gruppoid ). The term was then appropriated by B. A. Hausmann and Øystein Ore (1937)[1] in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is “perhaps most often used in modern mathematics” in thesensegiven to it in categorytheory.[2] According to Bergman and Hausknecht (1996): “There is no generally acceptedword for a set with a not necessarily associative binary operation. The word groupoid is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean “category in which all morphisms are invertible”. The term magma was used by Serre [Lie Algebras and Lie Groups, 1965].” [3] It also appears in Bourbaki's Éléments de mathématique , Algèbre, chapitres 1 à 3, 1970. [4]
If•isinsteada partial operation, then S is called a partial magma[5] or more often a partial groupoid.[5][6]
11.3 Morphism of magmas A morphism of magmas is a function, f : M → N , mapping magma M to magma N , that preserves the binary operation: f ( x •M y) = f ( x ) •N f (y)
where •M and •N denote the binary operation on M and N respectively.
11.4
Notation and combinatorics
The magma operation may be applied repeatedly, and in the general, non-associative case, the order matters, which is notated with parentheses. Also, the operation, •, is often omitted and notated by juxtaposition: (a • (b • c )) • d = (a(bc ))d
A shorthand is often used to reduce the number of parentheses, in which the innermost operations and pairs of 11.2 Definition parentheses are omitted, being replaced just with juxtaposition, xy • z = ( x • y ) • z . For example, the above is A magma is a set M matched with an operation, •, that abbreviated to the following expression, still containing sends any two elements a, b ∈ M to another element, a • parentheses: 64
65
11.7. CLASSIFICATION BY PROPERTIES
(a • bc )d .
Magma
A way to avoid completely theuse of parenthesesis prefix divisibility associativity notation, in which the same expression would be written ••a•bcd . Quasigroup Semigroup Thesetofallpossible strings consisting of symbols denoting elements of the magma, and sets of balanced parenidentity identity theses is called the Dyck language. The total number of different ways of writing n applications of the magma operator is given by the Catalan number, Cn . Thus, for Loop Monoid example, C 2 = 2, which is just the statement that ( ab)c and a(bc ) are the only two ways of pairing three elements associativity invertibility of a magma with two operations. Less trivially, C 3 = 5: ((ab)c )d , (a(bc ))d , (ab)(cd ), a((bc )d ), and a(b(cd )). Group The number of non-isomorphic magmas having 0, 1, 2, 3, 4, ... elements are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in OEIS). The corresponding numbers of non-isomorphic and non-antiisomorphic magmas Semilattices Semigroups where the operation is are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in commutative and idempotent OEIS).[7] Monoids Semigroups with identity elements
11.5
Free magma
A free magma, MX , onaset, X , is the “most general possible” magma generated by X (i.e., there are no relations or axioms imposed on the generators; see free object). It can be described as the set of non-associative words on X with parentheses retained:. [8] It can also be viewed, in terms familiar in computer science, as the magma of binary trees with leaves labelled by elements of X . The operation is that of joining trees at the root. It therefore has a foundational role in syntax. A free magma has the universal property such that, if f : X → N is a function from X to any magma, N , then there is a unique extension of f to a morphism of magmas, f ′
Groups Monoids
with inverse elements, or equivalently, associative loops or non-empty associative quasigroups
Abelian groups Groups where the operation is commu-
tative
Note that each of divisibility and invertibility imply the cancellation property.
11.7
Classification by properties
A magma (S , •), with x , y, u, z ∈ S , is called Medial If it satisfies the identity, xy • uz ≡ xu • yz
f ′ : MX → N .
Left semimedial
If it satisfies the identity, xx • yz ≡ xy
• xz See also: Free semigroup, Free group, Hall set, and Wedderburn–Etherington number Right semimedial If it satisfies the identity, yz • xx ≡ yx • zx Semimedial If it is both left and right semimedial
11.6
Types of magmas
Left distributive If it satisfies the identity, x • yz ≡ xy •
xz
Magmas are not often studied as such; instead there are several different kinds of magmas, depending on what Right distributive If it satisfies the identity, yz • x ≡ yx • zx axioms one might require of the operation. Commonly studied types of magmas include: Autodistributive If it is both left and right distributive Quasigroups Magmas Loops Quasigroups
where division is always possible Commutative If it satisfies the identity, xy ≡ yx
with identity elements
Semigroups Magmas where the operation is associative
Idempotent If it satisfies the identity, xx ≡ x Unipotent If it satisfies the identity, xx ≡ yy
66
CHAPTER 11. MAGMA
Zeropotent If it satisfies the identities, xx • y ≡ xx ≡ y •
xx [9]
Alternative If it satisfies the identities xx • y ≡ x • xy and
x • yy ≡ xy • y
Power-associative If
the submagma generated by any element is associative
A semigroup, or associative If
x • yz ≡ xy • z
it satisfies the identity,
A left unar If it satisfies the identity, xy ≡ xz A right unar If it satisfies the identity, yx ≡ zx
11.10
References
[1] Hausmann, B. A.; Ore, Øystein (October 1937), “Theory of quasi-groups”, American Journal of Mathematics 59 (4): 983–1004, doi:10.2307/2371362, JSTOR 2371362 [2] Hollings, Christopher (2014), Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups, American Mathematical Society, pp. 142–3, ISBN 9781-4704-1493-1 [3] Bergman, George M.; Hausknecht, Adam O. (1996), Cogroups and Co-rings in Categories of Associative Rings , American Mathematical Society, p. 61, ISBN 978-08218-0495-7
Semigroup with zero multiplication, or null semigroup [4] Bourbaki, N. (1998) [1970], “Algebraic Structures: §1.1 Laws of Composition: Definition 1”, Algebra I: Chapters If it satisfies the identity, xy ≡ uv 1–3, Springer, p. 1, ISBN 978-3-540-64243-5 Unital If it has an identity element Left-cancellative If, for all x , y , and, z, xy = xz implies
y = z
Right-cancellative If, for all x , y, and, z, yx = zx implies
y = z
Cancellative If
cancellative
it is both right-cancellative and left-
A semigroup with left zeros If
it is a semigroup and, for all x , the identity, x ≡ xy, holds If it is a semigroup and, for all x , the identity, x ≡ yx , holds
A semigroup with right zeros
If any triple of (not necessarily distinct) elements generates a medial submagma
Trimedial
[5] Müller-Hoissen, Folkert; Pallo, Jean Marcel; Stasheff, Jim, eds. (2012), Associahedra, Tamari Lattices and Related Structures: Tamari Memorial Festschrift , Springer, p. 11, ISBN 978-3-0348-0405-9
[6] Evseev, A. E. (1988), “A survey of partial groupoids”, in Silver, Ben, Nineteen Papers on Algebraic Semigroups , American Mathematical Society, ISBN 0-8218-3115-1 [7] Weisstein, Eric W., “Groupoid”, MathWorld . [8] Rowen, Louis Halle (2008), “Definition 21B.1.”, Graduate Algebra: Noncommutative View, Graduate Studies in Mathematics, American Mathematical Society, p. 321, ISBN 0-8218-8408-5 [9] Kepka, T.; Němec, P. (1996), “Simple balanced groupoids” (PDF), Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 35 (1): 53–60
Entropic If
it is a homomorphic image of a medial [10] Ježek, Jaroslav; Kepka, Tomáš (1981), “Free entropic cancellation magma.[10] groupoids” (PDF), Commentationes Mathematicae Universitatis Carolinae 22 (2): 223–233, MR 620359.
11.8 Generalizations See n-ary group.
• M. Hazewinkel (2001), “Magma”, in Hazewinkel,
Michiel, Encyclopedia of Mathematics , Springer, ISBN 978-1-55608-010-4
• M. Hazewinkel (2001), “Groupoid”, in Hazewinkel, 11.9 See also
• Magma category • Auto magma object • Universal algebra • Magma computer algebra system, named after the object of this article.
• Commutative non-associative magmas • Algebraic structures whose axioms are all identities • Groupoid algebra
Michiel, Encyclopedia of Mathematics , Springer, ISBN 978-1-55608-010-4
• M.
Hazewinkel (2001), “Free magma”, in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W., “Groupoid”, MathWorld . 11.11
Further reading
• Bruck, Richard Hubert (1971), A survey of bi-
nary systems (3rd ed.), Springer, ISBN 978-0-387-
03497-3
Chapter 12
Order This article is about order in group theory. For other If the order of group G is 1, then the group is called a uses in mathematics, see Order (mathematics). For other trivial group. Given an element a, ord(a) = 1 if and only uses, see Order. if a is the identity. If every (non-identity) element in G is the same as its inverse (so that a 2 = e ), then ord(a) consequently G is abelian since ab = (ab)−1 = In group theory, a branch of mathematics, the term order =−21 and b a−1 = ba by Elementary group theory. The converse is used in two unrelated senses: of this statement is not true; for example, the (additive) group Z 6 of integers modulo 6 is abelian, but the • The order of a group is its cardinality, i.e., the num- cyclic number 2 has order 3: ber of elements in its set. Also, the order, sometimes period,ofan element a ofagroupisthesmallest positive integer m such that am = e (where e denotes the identity element of the group, and am de- 2 + 2 + 2 = 6 ≡ 0 (mod 6) notes the product of m copies of a ). If no such m The relationship between the two concepts of order is the exists, a is said to have infinite order. following: if we write • The ordering relation of a partially or totally ordered group.
⟨a⟩ = {a
k
: k
∈ Z}
This article is about the first sense of order. for the subgroup generated by a, then The order of a group G is denoted by ord( G ) or | G | and the order of an element a is denoted by ord( a) or | a |. ord(a) = ord(⟨a⟩).
12.1
Example
For any integer k , we have ak = e if and only if ord(a) divides k .
The symmetric group S 3 has the following multiplication table. Example.
In general, the order of any subgroup of G divides the order of G . More precisely: if H is a subgroup of G , then ord(G ) / ord(H ) = [G : H ], where [G : H ] is called the index of H in G , an integer. This is Lagrange’s theorem. (This is, however, only truewhenGhasfiniteorder. Iford(G ) =∞,the quotient ord(G ) /ord(H ) does not make sense.)
This group hassix elements, so ord(S3 )=6. Bydefinition, theorder of theidentity, e, is1. Each of s, t , and w squares to e , so these group elements have order 2. Completing the enumeration, both u and v have order 3, for u 2 = v and u3 = vu = e, and v2 = u and v3 = uv = e.
As an immediate consequence of the above, we see that the order of every element of a group divides the order of 12.2 Order and structure the group. For example, in the symmetric group shown The order of a group and that of an element tend to speak above, where ord(S3 ) = 6, the orders of the elements are about the structure of the group. Roughly speaking, the 1, 2, or 3. more complicated the factorization of the order the more The following partial converse is true for finite groups: if complicated the group. d divides the order of a group G and d is a prime number, 67
68
CHAPTER 12. ORDER
then there exists an element of order d in G (this is sometimes called Cauchy’s theorem). The statement does not hold for composite orders, e.g. the Klein four-group does not have an element of order four). This can be shown by inductive proof.[1] The consequences of the theorem include: the order of a group G is a power of a prime p if and only if ord(a) is some power of p for every a in G .[2] If a has infinite order, then all powers of a have infinite order as well. If a has finite order, we have the following formula for the order of the powers of a: ord(ak ) = ord(a) / gcd(ord(a), k ) for every integer k . In particular, a and its inverse a −1 have the same order. In any group,
be used to prove that there are no (injective) homomorphisms between two concretely given groups. (For example, there can be no nontrivial homomorphism h : S3 → Z5 , because every number except zero in Z5 has order 5, which does not divide the orders 1, 2, and 3 of elements in S3.) A further consequence is that conjugate elements have the same order.
12.5
Class equation
An important result about orders is the class equation; it relates the order of a finite group G to the order of its center Z(G ) and the sizes of its non-trivial conjugacy classes:
|G| = |Z (G)| + ord(ab) = ord(ba)
�
di
i
where the di are the sizes of the non-trivial conjugacy classes; these are proper divisors of | G | bigger than one, and they are also equal to the indices of the centralizers in G of the representatives of the non-trivial conjugacy classes. For example, the center of S 3 is just the trivial group with the single element e , and the equation reads |S3 | = 1+2+3.
There is no general formula relating the order of a product ab to the orders of a and b. In fact, it is possible that both a and b have finite order while ab has infinite order, or that both a and b have infinite order while ab has finite order. An example of the former is a(x) = 2-x , b(x) = 1-x with ab(x) = x-1 in the group Sym(Z) . An example of the latter is a(x) = x+1, b(x) = x-1 with ab(x) = id . If ab = ba, we can at least say that ord( ab) divides lcm(ord(a), ord(b)). As a consequence, one can prove that in a finite 12.6 Open questions abelian group, if m denotes the maximum of all the orders of the group’s elements, then every element’s order Several deep questions about the orders of groups and divides m. their elements arecontainedin thevarious Burnside problems; some of these questions are still open.
12.3
Counting by order of elements
12.7 References
Suppose G is a finite group of order n,and d is a divisor of n. The number of order-d -elements in G is a multiple of [1] Conrad, Keith. “Proof of Cauchy’s Theorem” (PDF). Retrieved May 14, 2011. φ(d ) (possibly zero), where φ is Euler’s totient function, giving the number of positive integers no larger than d [2] Conrad, Keith. “Consequences of Cauchy’s Theorem” and coprime to it. For example, in the case of S 3 , φ(3) (PDF). Retrieved May 14, 2011. = 2, and we have exactly two elements of order 3. The theorem provides no useful information about elements of order 2, because φ(2) = 1, and is only of limited utility 12.8 See also for composite d such as d =6, since φ(6)=2, and there are zero elements of order 6 in S 3 . • Torsion subgroup
12.4
In relation phisms
to
homomor-
Group homomorphisms tend to reduce the orders of elements: if f : G → H is a homomorphism, and a is an element of G of finite order, then ord( f (a)) divides ord(a). If f is injective, then ord( f (a)) = ord(a). This can often
• Lagrange’s theorem (group theory)
Chapter 13
Ring This article is about an algebraic structure. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. In mathematics, a ring is one of the fundamen-
rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they also proved to be useful in other branches of mathematics such as geometry and mathematical analysis. A ring is an abelian group with a second binary operation that is associative, is distributive over the abelian group operation, andhasan identity element. By extension from the integers, theabelian group operation is called addition and the second binary operation is called multiplication . Whether a ring is commutative or not (i.e., whether the order in which two elements are multiplied changes or not the result) has profound implications on its behavior as an abstract object. As a result, commutative ring theory, commonly known as commutative algebra, is a key topic in ring theory. Its development has been greatly influenced by problems and ideas occurring naturally in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with the addition and multiplication of functions, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, and the cohomology ring of a topological space in topology.
Chapter IX of David Hilbert 's Die Theorie der algebraischen Zahlkörper. The chapter title is Die Zahlringe des Körpers, literally “the number rings of the field”. The word “ring” is the contraction of “Zahlring”.
13.1 tal algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions. The conceptualization of rings started in the 1870s and completed in the 1920s. Key contributors include Dedekind, Hilbert, Fraenkel, and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and
Definition and illustration
The most familiar example of a ring is the set of all integers, Z, consisting of the numbers . . . , −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, . . . The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings.
69
70
CHAPTER 13. RING
13.1.1
Definition
algebraic geometry often adopt the convention that “ring” means “commutative ring”, to simplify terminology. A ring is a set R equipped with binary operations[1] +and The additive group of a ring is the ring equipped just · satisfying the following three sets of axioms, called the with the structure of addition. Although the definition ring axioms[2][3][4] assumes that the additive group is abelian, this can be in1. R is an abelian group under addition, meaning that ferred from the other ring axioms.[6]
• (a + b) + c = a + (b + c ) for all a, b, c in
13.1.3 Basic properties
• a + b = b + a for all a, b in R (+ is
Some basic properties of a ring follow immediately from the axioms:
R (+ is associative).
commutative). • There is an element 0 in R such that a + 0 = a for all a in R (0 is the additive identity). • For each a in R there exists −a in R such that a + (−a) = 0 (−a is the additive inverse of a). 2. R is a monoid under multiplication, meaning that:
• (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c ) for all a, b, c in R (⋅ •
is associative). There is an element 1 in R such that a ⋅ 1 = a and 1 ⋅ a = a for all a in R (1 is the multiplicative identity).[5]
3. Multiplication is distributive with respect to addition:
• a ⋅ (b + c ) = (a ⋅ b) + (a ⋅ c ) for all a, b, c •
in R (left distributivity). (b + c ) ⋅ a = (b ⋅ a) + (c ⋅ a) for all a, b, c in R (right distributivity).
13.1.2 Notes on the definition
As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity. This article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except possibly the existence of a multiplicative identity 1 is called a rng (or sometimes pseudo-ring). For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. The operations + and ⋅ are called addition and multiplication, respectively. The multiplication symbol ⋅ is often omitted, so the juxtaposition of ring elements is interpreted as multiplication. For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not necessarily equal ba . Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or
• The additive identity, the additive inverse of each element, and the multiplicative identity are unique.
• For any element x in a ring R, one has x 0 = 0 = 0 x and (–1) x = – x .
• If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring.
• The binomial formula holds for any commuting pair of elements (i.e., any x and y such that xy = yx ).
13.1.4 Example: Integers modulo 4
See also: Modular arithmetic Equip the set Z4 = {0, 1, 2, 3} with the following operations:
• The sum x + y in Z4 is the remainder when the in-
teger x + y is divided by 4. For example, 2 + 3 = 1 and 3 + 3 = 2 .
• The product x · y in Z4 is the remainder when the integer xy is divided by 4. For example, 2 · 3 = 2 and 3 · 3 = 1 . Then Z 4 is a ring: each axiom follows from the corresponding axiom for Z. If x is an integer, the remainder of x when divided by 4 is an element of Z4 , and this element is often denoted by " x mod 4” or x , which is consistent with the notation for 0,1,2,3. The additive inverse of any x in Z4 is −x . For example, −3 = −3 = 1 . 13.1.5 Example: 2-by-2 matrices
Main article: Matrix ring The set of 2-by-2 matrices with real number entries is written
M2(R) =
�� � a c
b d
a,b,c,d
�
∈R
.
71
13.2. HISTORY
With the operations of matrix addition and matrix multi- 13.2.2 Hilbert plication, thisset satisfies the above ring axioms. The ele1 0 by David ment 0 1 is the multiplicative identity of the ring. If The term “Zahlring” (number ring) was coined [9] Hilbert in 1892 and published in 1897. In 19th century German, the word “Ring” could mean “association”, 0 1 0 1 0 0 A = and B = , then AB = which is still used today in English in a limited sense 1 0 0 0 0 1 (e.g., spy ring),[10] so if that were the etymology then it 1 0 while BA = 0 0 ; this example shows that the ring would be similar to the way “group” entered mathematics by being a non-technical word for “collection of reis noncommutative. More generally, for any ring R, commutative or not, and lated things”. According to Harvey Cohn, Hilbert used of “circling diany nonnegative integer n , one may form the ring of n - the term for a ring that had the property [11] Specifically, in rectly back” to an element of itself. by-n matrices with entries in R: see matrix ring. a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers “cycle back”. For instance, if a3 − 4a + 1 = 0 then a3 = 4a − 1, 13.2 History a4 = 4 a2 − a , a 5 = −a2 + 16 a − 4, a 6 = 16a2 − 8 a + 1, a7 = −8a2 + 65a − 16, and so on; in general, an is going See also: Ring theory § History to be an integral linear combination of 1, a, and a2 .
� � � � � � � �
� �
13.2.3
Fraenkel and Noether
The first axiomatic definition of a ring was given by Adolf Fraenkel in 1914,[12][13] but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse.[14] In 1921, Emmy Noether gave the modern axiomatic definition of (commutative) ring and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen .[15] 13.2.4 Multiplicative identity: mandatory vs. optional
Richard Dedekind , one of the founders of ring theory.
13.2.1
Dedekind
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers.[7] In 1871, Richard Dedekind defined the concept of the ring of integers of a number field.[8] In this context, heintroduced the terms “ideal” (inspired by Ernst Kummer's notion of ideal number) and “module” and studied their properties. But Dedekind did not use the term “ring” and did not define the concept of a ring in a general setting.
Fraenkel required a ring to have a multiplicative identity 1,[16] whereas Noether did not. [15] Most or all books on algebra[17][18] up to around 1960 followed Noether’s convention of not requiring a 1. Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin, [19] Atiyah and MacDonald,[20] Bourbaki,[21] Eisenbud, [22] and Lang.[23] But even today, there remain many books that do not require a 1. Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms. In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite directsums of rings, and the fact that properdirect summands of rings are not subrings. They conclude that “in many, maybe most, branches of ring theory the requirement of theexistence of a unity elementis not sensible, and therefore unacceptable.” [24]
72
CHAPTER 13. RING
In the second category, we find authors who use the following terms:[25][26]
• rings with multiplicative identity: unital ring, unitary ring , ring with unity , ring with identity, or ring with 1
• rings not requiring multiplicative identity: rng or pseudo-ring.
13.3 Basic examples Commutative rings :
• The prototype example is the ring of integers with the two operations of addition and multiplication.
• Therational, real andcomplexnumbers arecommutative rings of a type called fields.
• An algebra over a ring is itself a ring. These are also modules. Some examples:
• Any algebra over a field. • The polynomial ring R[X ] of polynomials over
• The set of all continuous real-valued functions defined on the real line forms a commutative ring. The operations are pointwise addition and multiplication of functions.
• Let X be a set and R a ring. Then the set of all func-
tions from X to R forms a ring, which is commutative if R is commutative. The ring of continuous functions in the previous example is a subring of this ring if X is the real line and R is the field of real numbers.
Noncommutative rings:
• For any ring R and any natural number n , the set
of all square n -by-n matrices with entries from R , forms a ring with matrix addition and matrix multiplication as operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G ) of G .
The operations in this ring are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by End R(V ).
a ring R is itself a ring. A free module over R of infinite dimension • If G is a group and R is a ring, the group ring of • Z[c] , the integers with an irrational number c G over R is a free module over R having G as baadjoined. A free module of infinite dimension sis. Multiplication is defined by the rules that the if c is a transcendental number, a free module elements of G commute with the elements of R and of finite dimension if c is an algebraic integer multiply together as they do in the group G . • Z[1/n] , the set of fractions whose denomina- • Many rings that appear in analysis are noncommutorsareapowerof n (including negative ones). tative. For example, most Banach algebras are nonA non-free module. commutative. • Z[1/10]√ , the set of decimal fractions. • Z[(1+ d)/2] , where d is a square-free inte- Non-rings: ger of the form 4n+1. A free module of rank two. Cf. Quadratic integers. • The set of natural numbers N with the usual opera• Z[i] , the Gaussian integers. tions is not a ring, since (N, +) is not even a group √ (the elements are not all invertible with respect to • Z[(1 + −3)/2] , the Eisenstein integers. addition). For instance, there is no natural number Also their generalization, a Kummer ring. which can be added to 3 to get 0 as a result. There • The set of all algebraic integers forms a ring. This is a natural way to make it a ring by adding negafollows for example from the fact that it is the tive numbers to the set, thus obtaining the ring of integral closure of the ring of rational integers in the integers. The natural numbers (including 0) form field of complex numbers. The rings in the three an algebraic structure known as a semiring (which previous examples are subrings of this ring. has all of the properties of a ring except the additive inverse property). • The set of formal power series R[[X 1, …, Xn]] over a commutative ring R is a ring. • Let R be the set of all continuous functions on the real line that vanish outside a bounded interval de• If S is a set, then the power set of S becomes a ring pending on the function, with addition as usual but if we define addition to be the symmetric difference with multiplication defined as convolution: of sets and multiplication to be intersection. This ∞ corresponds to a ring of sets and is an example of a (f ∗ g )(x) = f (y )g(x − y )dy. Boolean ring. −∞
�
73
13.4. BASIC CONCEPTS
Then R is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.
13.4
Basic concepts
13.4.1
Elements in a ring
A left zero divisor of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0 .[27] A right zero divisor is defined similarly. A nilpotent element is an element a such that an = 0 for some n > 0 . One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor. An idempotent e is an element such that e 2 = e . One exampleof an idempotent element is a projectionin linear algebra. A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a−1 . The set of units of a ring is a group under ring multiplication; this group is denoted by R× or R∗ or U (R) . For example, if R is the ring of all square matrices of size n over a field, then R × consists of the set of all invertible matrices of size n, and is called the general linear group. 13.4.2
Subring
copies of 1 and −1 together many times in any mixture. It is possible that n · 1 = 1 + 1 + . . . + 1 (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the characteristic of R . In some rings, n · 1 is never zero for any positive integer n , and those rings are said to have characteristic zero . Given a ring R , let Z(R) denote the set of all elements x in R such that x commutes with every element in R: xy = yx for any y in R . Then Z(R) is a subring of R ; called the center of R . More generally, given a subset X of R , let S be the set of all elements in R that commute with every element in X . Then S is a subring of R, called the centralizer (or commutant) of X . The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they generate a subring of the center. 13.4.3 Ideal
Main article: Ideal (ring theory) The definition of an ideal in a ring is analogous to that of normal subgroup in a group. But, in actuality, it plays a role of an idealized generalization of an element in a ring; hence, thename “ideal”. Like elements of rings, thestudy of ideals is central to structural understanding of a ring. Let R be a ring. A nonempty subset I of R is then said to be a left ideal in R if, for any x , y in I and r in R, x + y and rx are in I . If RI denotes the span of I over R; i.e., the set of finite sums
Main article: Subring A subset S of R is said to be a subring if it can be regarded as a ring with the addition and the multiplication restricted from R to S . Equivalently, S is a subring if it is not empty, and for any x , y in S , xy , x + y and −x are in S . If all rings have been assumed, by convention, to have a multiplicative identity, then to be a subring one would also require S to share the same identity element as R.[28] So if all rings have been assumed to have a multiplicative identity, then a proper ideal is not a subring. For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z [X ] (in both cases, Z contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers 2 Z does not contain the identity element 1 and thus does not qualify as a subring. An intersection of subringsis a subring. Thesmallest subring containing a given subset E of R is called a subring generated by E . Such a subring exists since it is the intersection of all subrings containing E . Foraring R,thesmallestsubringcontaining1iscalledthe characteristic subring of R. It can be obtained by adding
r1 x1 +
··· + r
∈ R, x ∈ I, then I is a left ideal if RI ⊆ I . Similarly, I is said to be right ideal if IR ⊆ I . A subset I is said to be a n xn ,
ri
i
two-sided ideal or simply ideal if it is both a left ideal
and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then RE is a left ideal, called the left ideal generated by E ; it is the smallest left ideal containing E . Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R. If x isin R,then Rx and xR areleftideals andrightideals, respectively; they are called the principal left ideals and right ideals generated by x . The principal ideal RxR is written as ( x) . For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be a simple if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no
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CHAPTER 13. RING
strictly increasing infinite chain of left ideals is called a left Noetherian ring. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left Artinian ring. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the Hopkins–Levitzki theorem). The integers, however, form a Noetherian ring which is not Artinian. For commutative rings, the ideals generalize the classical notion of divisibility anddecomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a prime ideal if for any elements x, y ∈ R we have that xy ∈ P implies either x ∈ P or y ∈ P . Equivalently, P is prime if for any ideals I , J we have that I J ⊆ P implies either I ⊆ P or J ⊆ P. This latter formulation illustrates the idea of ideals as generalizations of elements.
• The Galois group of a field extension L/K is the set
of all automorphisms of L whose restrictions to K are the identity.
• For any ring R, there are a unique ring homomorphism Z → R and a unique ring homomorphism R →0.
• An epimorphism (i.e., right-cancelable morphism) of rings need not be surjective. For example, the unique map Z → Q is an epimorphism. • An algebra homomorphism from a k -algebra to the endomorphism algebra of a vector space over k is called a representation of the algebra.
13.4.4 Homomorphism
Given a ring homomorphism f : R → S , the set of all elements mapped to 0 by f is called the kernel of f . Main article: Ring homomorphism The kernel is a two-sided ideal of R. The image of f , on the other hand, is not always an ideal, but it is always a A homomorphism from a ring (R, +, · ) to a ring (S , ‡, subring of S . *) is a function f from R to S that preserves the ring op- To give a ring homomorphism from a commutative ring erations; namely, such that, for all a, b in R the following R to a ring A with image contained in the center of A is identities hold: the same as to give a structure of an algebra over R to A (in particular gives a structure of A-module). • f (a + b) = f (a) ‡ f (b)
• f (a · b) = f (a) * f (b) • f (1R) = 1S If one is working with not necessarily unital rings, then the third condition is dropped. A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to f (i.e., a ring homomorphism which is an inverse function). Any bijective ring homomorphism is a ring isomorphism. Two rings R, S are said to be isomorphic if there is an isomorphism between them and in that case one writes R ≃ S . A ring homomorphism between the same ring is called an endomorphism and an isomorphism between the same ring an automorphism. Examples:
• The function that maps each integer x to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring Z to the quotient ring Z/4Z (“quotient ring” is defined below).
• If u is−1a unit element in a ring R, then R → R, x → is a ring homomorphism, called an inner automorphism of R. uxu
• Let R be a commutative ring of prime characteristic p. Then x → x is a ring endmorphism of R called p
the Frobenius homomorphism.
13.4.5
Quotient ring
Main article: Quotient ring The quotient ring of a ring, is analogous to the notion of a quotient group of a group. More formally, given a ring (R, +, · ) and a two-sided ideal I of (R, +, · ), the quotient ring (or factor ring ) R/I is the set of cosets of I (with respect to the additive group of (R, +, · ); i.e. cosets with respect to (R, +)) together with the operations: (a + I ) + (b + I ) = (a + b) + I and (a + I )(b + I ) = (ab) + I . for every a, b in R. Like the case of a quotient group, there is a canonical map p : R → R/I given by x → x + I . It is surjective and satisfies the universal property: if f : R → S is a ring homomorphism such that f (I ) = 0 , then there is a unique f : R /I → S such that f = f ◦ p . In particular, taking I to be the kernel, one sees that the quotient ring R/ ker f is isomorphic to the image of f ; the fact known as the first isomorphism theorem. The last fact implies that actually any surjective ring homomorphism satisfies the universal property since the image of such a map is a quotient ring.
75
13.6. CONSTR CONSTRUCTIONS UCTIONS
13.5 13.5
Ring Ring acti action on:: a mod modul ulee ove overr a section ring of ring of L . A particularly important case is when L is the canonical the canonical line bundle and bundle and then R is the canonical the canonical ring ring of the base variety. ring of
Main article: Module article: Module (mathematics) In group theo theory ry,, one can cons consid ider er the ac actition on of a gr grou oupp on a set. To give a group action, action, say, G acting acting on a set S , is to givea group homomorp homomorphism hism from G tothe automorphism group of group of S (that (that is, the symmetric the symmetric group of group of S .).) In much the same way, one can consider a ring action; that is, a ring a ring homomorphism f from a ring R to the endomorphism ring of ring of an abelian group M . One usually writes rm or r ⋅m for for f (r )m and calls M a left a left module over module over R. If R is a field, this amounts to giving a structure of a vector space on M . In par partic ticula ular, r, a ring ring R isaleftmoduleover R itself itself through through l :: R →End(R), l (r ) x x = rx (called (called the le left ft regu regular lar rep represe resenntation of tation of R ). Some ring-theoretic concepts can be stated in a modul module-t e-the heore oretiticc langua language: ge: for examp example le,, a subse subsett of a ring R isaleftidealof R ifandonlyifitisan R-submodule with respect to the left R -module structure of R . A left ideal is principal if and only if it is a cyclic a cyclic submodule. submodule . A Z-mod -modul ulee is the the same same thin thingg as an abe abelian lian grou groupp; this this allowsonetousethe modu module le theo theory ry to study study abe abelia liann groups. groups. For example, in general, if M is is a left module over a ring R that is cyclic; i.e., M = = Rx for for some x , then M is is isomorph morphic ic to the quoti quotient ent of R byth by thee kerne ernell of R → M , r ↦ ↦ rx . In particular, if if R is Z, then any cyclic any cyclic group (which group (which is cyclic as Z -module) is of the form Z /nZ, recovering the usual classification of cyclic groups. See § See § Domains for Domains for an example of an application to linear algebra. Any ring homomorphism induces the structure of a module: if f : R → S is is a ring homomorphism, then S is a left module over R by the formula: r ⋅s = f (r )s. A module that is also a ring is called an algebra an algebra over over the base ring (provided (provided the base ring is central). Example: Geometrically, Geometrically, a module can be viewed as an algebraic counterpart of a vector a vector bundle. bundle. Let E be be a vector bun bundle dle over over a compa compact ct spac space, e, and Γ(E )thespaceofits ) thespaceofits section sections. s. Then Then Γ(E )isamoduleoverthering ) isamoduleoverthering R of contincontinuous functions on the base space. Swan’s theorem states theorem states that, via Γ, the category of vector bundles is equivalent to the category of finitely generated projective generated projective R -modules (“projective” corresponds to local trivialization.) In application, one often cooks up a ring by summing up modules. Continuing the above geometric example, let L be a line a line bundle on bundle on an algebraic variety (Γ( L) is a module over the coordinate ring of the variety). Then the direct the direct sum of modules
13.6 Constr Construct uctio ions ns 13.6.1 13.6.1
Direct Direct produ product ct
Main article: Direct article: Direct product of rings Let R and S be rings. rings. Then Then the product the product R × S can be equipped with the following natural ring structure:
• (r 1, s1) + (r 2, s2) = (r 1 + r 2, s1 + s2) • (r 1, s1) ⋅ (r 2, s2) = (r 1 ⋅ r 2, s1 ⋅ s2) for for every every r 1 , r 2 in R and s1 , s2 in S .Thering . Thering R × S with with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the direct product of R with S . The same construction also works for an arbitrary family of rings: if Ri are rings indexed by a set I , then i∈I R i is a ring with componentwise addition and multiplication. Let R be a commutative commutative ring and a 1 ,··· ,a be ideals such ̸ = j . Then the Chinese that a i + aj = (1) whenever i̸ the Chinese remainder theorem says theorem says there is a canonical ring isomorphism:
∏
n
R / ( ai )
∩ ≃
� ∏
R/ai ,
x
→ (x mod a1, . . . , x mod a
n)
A “finite” direct product may also be viewed as a direct sum of ideals.[29] Namely, let Ri , 1 ≤ i ≤ n be rings, Ri → R = Ri the inclusions with the images a i (in particular a i are rings though not subrings). Then ai are ideals of R and R = a 1
⊕···⊕a
n,
ai aj = 0, i = j,
̸
a2i
⊆ a
i
as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearlythedirectsumofsuchidealsalsodefinesaproduct of rings that is isomorphic to R. Equivalently, the above can be done through t hrough central central idempotents. idempotents. Assume R has the above decomposition. Then we can write 1 = e 1 +
··· + e
n,
∈ a .
ei
i
By the conditions on ai , one has that ei are central idem̸ = j (orthogonal). potents and ei ej = 0, i̸ (orthogonal). Again, one one can reverse the construction. construction. Namely, Namely, if one is given a ⊕n≥0Γ(L⊗n) partition of 1 in orthogonal central idempotents, idempotents, then let has the structure of a commutative ring; it is called the ai = Re i , which are two-sided ideals. If each ei is not a
76
CHAPTER CHAPTER 13. RING RING
sum of orthogonal central idempotents,[30] then their direct sum is isomorphic to R. An important application of an infinite direct product is the construction of a projective a projective limit of limit of rings (see below). Another application is a restricted a restricted product of product of a family of rings (cf. adele (cf. adele ring). ring). 13.6.2 Polyn Polynomi omial al ring
Main article: Polynomial article: Polynomial ring Given a symbol t (called (called a variable) and a commutative ring R, the set of polynomials
there exists a unique ring homomorphism φ : R [t] → S such that φ(t) = x and φ restricts to φ .[31] For example, choosing a basis, a symmetric a symmetric algebra satisfies algebra satisfies the universal property and so is a polynomial ring. To give an example, let S be be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be be the identity function. function. Each r in in R defines defines a constant function, function, giving rise to the homomorR S → phism . The universal universal property says says that this map extends uniquely to R[t]
→ S, f → f
(t maps maps to x ) where f is is the polynomial the polynomial function defined function defined by f . The resulting resulting map is injective injective if and only if R is R[t] = an tn + an−1 tn−1 + · · · + a1 t + a0 | n ≥ 0, aj ∈infinite. R Given a non-constant monic polynomial f in in R [t] , there forms a commutative ring with the usual addition and exists a ring S containing R such that f is a product of multiplication, containing R as a subrin subring. g. It is calle calledd linear factors in S [t] .[32] the polynomial ring over R. More More gene general rally ly,, the set be an algebraically closed field. The Hilbert’s NullR[t1 , . . . , tn ] of all polynomials in variables t1 , . . . , t n Let k be stellensatz (theorem (theorem of zeros) states that there is a natural forms a commutative ring, containing R[ti ] as subrings. stellensatz one-to-one correspondence between the set of all prime If R is an integral domain, then R [t] is also an integral ideals in k [t , . . . , t ] and the set of closed subvarieties n 1 domain; its field of fractions is the field of rational of rational func- of k n . In particular, many local problems problems in algebraic algebraic tions.. If R is a noetherian ring, then R [t] is a noetherian tions noetherian geometry may be attacked through through the study of the genring. If R is a unique factorization factorization domain, then t hen R [t] is erators erators of an ideal ideal in a polynom polynomial ial ring. ring. (cf. Gröbner a unique factorization domain. Finally, R is a field if and basis basis.).) only if R[t] is a principal ideal domain. There There are some other related related constructi constructions. ons. A formal Let R ⊆ S be be commutative rings. Given an element x of of power series ring R[[t]] consists of formal power series S , one can consider the ring homomorphism
{
R[t]
→ S, f → f (x)
(i.e., the substitution substitution).). If S =R[t ]and ] and x =t , then f (t )= )= f . Because of this, the polynomial f is often also denoted by f (x) is denoted by f (t) . The image of the map f → R[x] ; it is the same thing as the subring of S generated generated by R and x . Example: k[t2 , t3] denotes the image of the homomorphism k [x, y ]
→ k[t], f → f (t2, t3).
In other words, it is the subalgebra of k[t] generated by t 2 and t 3 . Example: let f be be a polyno polynomi mial al in one varia variabl ble; e; i.e., i.e., an elelemen ementt in a polyn polynomi omial al ring ring R.Then f (x+h) is an elem elemen entt in R[h] and f (x + h) − f (x) is divisible by h in that ring. Theresult Theresult of subst substitu itutin tingg zero zero to h in (f (x+h)−f (x))/h ′ is f (x) , the derivative of f at at x . The substitution is a special case of the universal property of a polynomial polynomial ring. ring. The property property states: given a ring homomorphism φ : R → S and and an element x in in S
}
∞
�
ai ti ,
0
∈ R
ai
together with multiplication and addition that mimic those for convergent series. It contains R[t] as a subring. Note a formal power series ring does not have the universal property of a polynomial ring; a series may not converge after after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local is local (in (in fact, complete fact, complete).). 13.6. 13.6.3 3
Matr Matrix ix ring ring and and endo endomo morp rphi hism sm ring
Main articles: Matrix articles: Matrix ring and ring and Endomorphism Endomorphism ring Let R be a ring (not necessarily commuta commutative). tive). The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix usual matrix multiplication.. It is called the matrix plication the matrix ring and ring and is denoted by Mn(R). Given a right R-module U , , the set of all R-linear maps from U to to itself forms a ring with addition that is of function and multiplication that is of composition of composition of
77
13.6. CONSTR CONSTRUCTIONS UCTIONS
• The function functions; it is called the endomorphism ring of U and functions; and is The function field of an algebraic variety over variety over a field denoted by EndR (U ) . k islim k [U ] wher wheree the limit limit runs runs over over allthe all the coordi coordi-−→ k [U ] of nonempty open subsets U (more nate rings (more As in linear algebra, a matrix ring may be canonically succinctly it is the stalk the stalk of of the structure sheaf at the interpre interpreted ted as an endomorphi endomorphism sm ring: EndR (Rn ) ≃ generic point.) point.) Mn (R) . This is a special case of the following fact: If n f : ⊕ n 1 U → ⊕1 U is an R -linear map, then f may be Any commutative ring is the colimit of finitely of finitely generated written as a matrix with entries f ij ij in S = EndR (U ) , subrings. subrings . resulting in the ring isomorphism: A pro projectiv jectivee limit (ora filte filtered red limi limitt)ofringsisdefinedas follows. Suppose we're given a family of rings Ri , i run runn EndR (⊕1 U ) → Mn (S ), f → (f ij ning over over positiv positivee integers integers,, say, and ring homomorp homomorphis hisms ms ij ). Rj → Ri , j ≥ i such that R i → Ri are all the identities Any ring homomorphism R → S induces Mn(R) → and R k → R j → Ri is Rk → Ri whenever k ≥ j ≥ i Mn(S );); in fact, fact, any ring homomorphism homomorphism between matrix . Then lim Ri is the subring of Ri consisting of (xn ) ←− rings arises in this way. [33] such that x j maps to x i under Rj → Ri , j ≥ i . Schur’s lemma says lemma says that if U is is a simple right R-module, For an example of a projective limit, see #completion see #completion.. r U i⊕m is then EndR (U ) is a division ring. [34] If U = i=1 13.6.5 Locali Localizat zatio ion n a direct sum of mi -copies -copies of simple R-modules U i , then 13.6.5
∏
⊕
i
r
EndR (U )
≃
⊕
Mm (EndR (U i )) i
1
The Artin–Wedderb The Artin–Wedderburn urn theorem theorem states states any semisimple any semisimple ring (cf. ring (cf. below) below) is of this form. A ring R and the matrix ring Mn(R) over it are Morita are Morita equivalent:: the category of right modules of R is equivequivalent alent to the category of right modules over M n(R).[33] In particular, two-sided ideals ideals in R corresp correspond ond in one-to-on one-to-onee to two-sided ideals in M n(R). Examples:
• The autom automorp orphis hisms ms of the pr proojec jectitive ve lin linee ov over er a rin ringg are given by homographies by homographies from from the 2 x 2 matrix ring.
13.6.4 Limits Limits and colimi colimits ts of rings rings
Let Ri be be a sequence of rings such that Ri is is a subring of Ri ₊₁ ₊₁ for all i . Then the union (or filtered (or filtered colimit) colimit ) of Ri is is Ri defined as follows: it is the disjoint union the ring lim −→ of all Ri 's's modulo the equivalence relation x ∼ y if and only if x = y in Ri for for sufficiently large i . Examples of colimits:
• A
polynom polynomial ial ring in infinite infinitely ly many many variabl variables: es: R[t1 , t2 , · · · , tm ]. · · · ] = lim −→
R[t1 , t2 ,
• The algebraic The algebraic closure of finite of finite fields of fields of the same characteristic F = lim −→ F . • The field of formal Laurent series over series over a field k : − k ((t)) = lim t k[[t]] (it is the field of fractions of −→ the formal the formal power series ring k [[t]] .) p
m
The localization The localization generalizes generalizes the construction construction of the field the field of fractions of fractions of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring R and a subset S of of R, there exists a ring R [S −1 ] together with the ring homomor homomorphis phism m R → R[S −1] that “invert “inverts” s” S ; that is, the homomorphism maps elements in S to to unit 1 − elements in R[S ] , and, moreover, any ring homomorphism from R that “inverts” S uniquely factors through R[S −1 ] . [35] The ring R [S −1 ] is called the localization of R with with resp respec ectt to S . For exampl example, e, if R is a commutativ commutativee ring and f an an element in R, then the localization R [f −1 ] consists of elements of the form r/f n , r ∈ R, n ≥ 0 (to − 1). )[36] be precise, R[f −1 ] = R [t]/(tf − The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal (or a union of prime ideals) in R. In that case S = R − p , one often writes Rp for R[S −1 ] . Rp is then a local ring with the maximal ideal p Rp . This is the reason for the terminology “localization”. “localization”. The field of fractions fractions of an integral domain R is the localization of R at the prime ideal ideal zero. zero. If p is a prime ideal of a commutative ring R, then the field of fractions of R /p is the same as the residue field of the local ring Rp and is denoted by k(p) . If M is is a left R -module, then the localization of M with with 1 − respect to S is is given by a change a change of rings M [S ] = R[S −1 ] ⊗R M . The most important properties of localization are the following: when R is a commutative ring and S a a multiplicatively closed subset a bijection between the set of all prime ideals in R disjoint from S and and the set of all − [37] 1 prime ideals in R [S ] . R[S −1 ] = lim R[f −1 ] , f running running over elements in − → S with with partial ordering given by divisibility.[38]
•p→
pm
•
p[S −1 ] is
78
CHAPTER 13. RING
• The localization is exact: 0 → M ′ [S −1 ] → M [S −1 ] → M ′′ [S −1 ] → 0 is exact over R[S −1 ] whenever 0 → M ′ → M → M ′′ → 0
A complete ring has much simpler structure than a commutative ring. This owns to the Cohen structure theorem, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the integral clois exact over R. sure and completion has been among the most important that distinguish modern commutative ring theory • Conversely, if 0 → M m′ → M m → M m′′ → ′0 is aspects exact for any maximal ideal m , then 0 → M → from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexM → M ′′ → 0 is exact. amination of the roles of Noetherian rings and motivated, • A remark: localization is no help in proving a global among other things, the definition of excellent ring. existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this 13.6.7 Rings with generators and relations is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently The most general way to construct a ring is by specifya local notion.) ing generators and relations. Let F be a free ring (i.e., free algebra over the integers) with the set X of symbols; In category theory, a localization of a category amounts i.e., F consists of polynomials with integral coefficients to making some morphisms isomorphisms. An element in noncommuting variables that are elements of X . A free in a commutative ring R may be thought of as an endo- ring satisfies the universal property: any function from morphism of any R-module. Thus, categorically, a local- the set X to a ring R factors through F so that F → R ization of R with respect to a subset S of R is a functor is the unique ring homomorphism. Just as in the group every ring can be represented as a quotient of a free from the category of R -modules to itself that sends ele- case,[40] ments of S viewed as endomorphisms to automorphisms ring. and is universal with respect to this property. (Of course, Now, we can impose relations among symbols in X by R then maps to R [S −1 ] and R -modules map to R [S −1 ] taking a quotient. Explicitly, if E is a subset of F , thenthe -modules.) quotientring of F by theideal generated by E is called the ring with generators X and relations E . If we used a ring, say, A as a base ring instead of Z, then the resulting ring 13.6.6 Completion will be over A. For example, if E = {xy − yx | x, y ∈ X } , then the resulting ring will be the usual polynomial Let R be a commutative ring, and let I be an ideal of R . ring with coefficients in A in variables that are elements The completion of R at I is the projective limit Rˆ = of X (It is also the same thing as the symmetric algebra R/I n ; it is a commutative ring. The canonical homo- over A with symbols X .) lim ← − morphisms from R to the quotients R/I n induce a homo morphism R → Rˆ . The latter homomorphism is injec- In the category-theoretic terms, the formation S → tive if R is a noetherian integral domain and I is a proper set the by generated ring free the S is the left adjoint ideal, or if R is a noetherian local ring with maximal ideal functor of the forgetful functor from the category of rings I , by Krull’s intersection theorem.[39] The construction is to Set (and it is often called the free ring functor.) especially useful when I is a maximal ideal. Let A, B be algebras over a commutative ring R. Then the The basic example is the completion Z p of Z at the prin- tensor product of R-modules A ⊗R B is a R-module. We cipal ideal ( p) generated by a prime number p; it is called canturnittoaringbyextendinglinearly (x ⊗ u)(y ⊗ v) = the ring of p -adic integers. The completion can in this xy ⊗ uv . See also: tensor product of algebras, change of case be constructed also fromthe p-adic absolute value on rings. Q.The p-adic absolute value on Q isamap x → |x| from Q to R given by |n| p = p −v (n) where v p (n) denotes the exponent of p in the prime factorization of a nonzero in- 13.7 Special kinds of rings teger n into prime numbers (we also put |0| p = 0 and |m/n| p = |m| p/|n| p ). It defines a distance function on Q and the completion of Q as a metric space is denoted 13.7.1 Domains by Q p. It is again a field since the field operations extend to the completion. The subring of Q p consisting of A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral doelements x with |x| p ≤ 1 is isomorphic to Z p. main. The most important integral domains are princiSimilarly, the formal power series ring R [[t]] is the com- pal ideals domains, PID for short, and fields. A principal pletion of R [t] at (t) . ideal domain is an integral domain in which every ideal See also: Hensel’s lemma. is principal. An important class of integral domains that p
79
13.7. SPECIAL KINDS OF RINGS
contain a PID is a unique factorization domain (UFD), an integral domain in which everynonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an “ideal” admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.[41] Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial q. Then, since k[t] is a unique factorization domain, q factors into powers of distinct irreducible polynomials (i.e., prime elements): q = p e11 ...pess .
Letting t · v = f (v ) , we make V a k [t ]-module. The structure theorem then says V is a direct sum of cyclic modules, each of which is isomorphic to the module of the form k[t]/( pki ) . Now, if pi (t) = t − λi , then such a cyclic module (for pi ) has a basis in which the restriction of f is represented by a Jordan matrix. Thus, if, say, k is algebraically closed, then all p i 's are of the form t − λi and the above decomposition corresponds to the Jordan canonical form of f . In algebraic geometry, UFD’s arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a regular local ring. A regular local ring is a UFD. [42] The following is a chain of class inclusions that describes the relationship between rings, domains and fields: j
rings ⊃ integral domains ⊃ integrally closed domains ⊃unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields
• Commutative
13.7.2
Division ring
A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominentexampleofadivisionringthatisnotafieldisthering of quaternions. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turnedout that every finite domain (in particular finite division ring) is a field; in particular commutative (the Wedderburn’s little theorem). Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings. Cartan famously
asked the following question: given a division ring D and a proper sub-division-ring S that is not contained in the center, does each inner automorphism of D restrict to an automorphism of S ? The answer is negative: this is the Cartan–Brauer–Hua theorem. A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra. 13.7.3
Semisimple rings
A ring is called a semisimple ring if it is semisimple as a left module (or right module) over itself; i.e., a direct sum of simple modules. A ring is called a semiprimitive ring if its Jacobson radical is zero. (The Jacobson radical is the intersection of all maximal left ideals.) A ring is semisimple if and only if it is artinian and is semiprimitive. An algebra over a field k is artinian if and only if it has finite dimension. Thus, a semisimple algebra over a field is necessarily finite-dimensional, while a simple algebra may have infinite dimension; e.g., the ring of differential operators. Any module over a semisimple ring is semisimple. (Proof: any free module over a semisimple ring is clearly semisimple and any module is a quotient of a free module.) Examples of semisimple rings:
• A matrix ring over a division ring is semisimple (actually simple).
• The group ring k [G] of a finite group G over a field
k is semisimple if the characteristic of k does not divide the order of G . (Maschke’s theorem)
• The Weyl algebra (over a field) is a simple ring; it is not semisimple since it has infinite dimension and thus not artinian.
• Clifford algebras are semisimple. Semisimplicity is closely related to separability. An algebra A over a field k is said to be separable if the base extension A ⊗k F is semisimple for any field extension F /k . If A happens to be a field, then this is equivalent to the usual definition in field theory (cf. separable extension.) 13.7.4 Central simple algebra and Brauer group
Main article: Central simple algebra For a field k , a k -algebra is central if its center is k and is simpleifitisa simplering. Sincethecenterofasimple k algebra is a field, any simple k -algebra is a central simple
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CHAPTER 13. RING
algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a k algebra. The matrix ring of size n over a ring R will be denoted by Rn . The Skolem–Noether theorem states any automorphism of a central simple algebra is inner. Two central simple algebras A and B aresaidtobe similar if there are integers n and m such that A ⊗k k n ≈ B ⊗k km . [43] Since kn ⊗k km ≃ knm , the similarity is an equivalence relation. The similarity classes [ A] with the multiplication [A][B ] = [A ⊗k B ] form an abelian group called the Brauer group of k and is denoted by Br(k) . By the Artin–Wedderburn theorem, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, Br(k) is trivial if k isa finite fieldor an algebraically closed field (more generally quasi-algebraically closed field; cf. Tsen’s theorem). Br(R) has order 2 (a special case of the theorem of Frobenius). Finally, if k is a nonarchimedean local field (e.g., Q p ), then Br(k) = Q/Z through the invariant map. Now, if F is a field extension of k , then the base extension −⊗k F inducesBr(k) → Br(F ) . Itskernelisdenotedby Br(F /k) . It consists of [A] such that A ⊗k F is a matrix ring over F (i.e., A is split by F .) If the extension is finite and Galois, then Br (F /k) is canonically isomorphic to H 2 (Gal(F /k ), k∗ ) .[44] Azumaya algebras generalize the notion of central simple algebras to a commutative local ring. 13.7.5
convolution: (f g )(t) =
∗
�
f (s)g (t
s G
∈
− s)
It also comes with thevaluation v such that v( f ) is the least element in the support of f . The subring consisting of elements with finite support is called the group ring of G (which makes sense even if G is not commutative). If G is the ring of integers, then we recover the previous example (by identifying f with theseries whose n-th coefficient is f (n).) See also: Novikov ring and uniserial ring.
13.8 Rings with extra structure A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:
• An associative algebra is a ring that is also a vector
space over a field K such that the scalar multiplication distributes over the ring multiplication. For instance, the set of n-by-n matrices over the real field R has dimension n2 as a real vector space.
• A ring R is a topological ring if its set of elements R is given a topology which makes the addition map ( + : R × R → R ) and the multiplication map ( · : R × R → R ) to be both continuous as maps be-
Valuation ring
Main article: valuation ring If K is a field, a valuation v is a group homomorphism from the multiplicative group K * to a totally ordered abelian group G such that, for any f , g in K with f + g nonzero, v ( f + g ) ≥ min{v( f ), v( g)}. The valuation ring of v is the subring of K consisting of zero and all nonzero f such that v( f ) ≥ 0. Examples:
tween topological spaces (where X × X inherits the product topology or any other product in the category). For example, n -by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.
• A λ-ring isn a commutative ring R together with op-
erations λ : R → R that arelike n-th exterior powers:
• Thefieldof formal Laurent series k((t)) over a field k comes with the valuation v such that v( f ) is the least degree of a nonzero term in f ; the valuation ring of v is the formal power series ring k [[t]] .
n
n
λ (x + y ) =
�
λi (x)λn−i (y )
0
• More generally, given a field
k and a totally or-
dered abelian group G , let k((G)) be the set of all functions from G to k whose supports (the sets of points at which the functions are nonzero) are well ordered. It is a field with the multiplication given by
For example, Z is a λ-ring with λ n (x) = nx , the binomial coefficients. The notion plays a central rule in the algebraic approach to the Riemann–Roch theorem.
81
13.10. CATEGORY THEORETICAL DESCRIPTION
13.9
Some examples of the ubiquity 13.9.3 Representation ring of a group ring of rings To any group ring or Hopf algebra is associated its
representation ring or “Green ring”. The representation Many different kinds of mathematical objects can be ring’s additive group is the free abelian group whose bafruitfully analyzed in terms of some associated ring. sis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decom13.9.1 Cohomology ring of a topological position of the module. The multiplication is the tensor product. When the algebra is semisimple, the representaspace tion ring is just the character ring from character theory, To any topological space X one can associate its integral which is more or less the Grothendieck group given a ring structure. cohomology ring
H ∗ (X, Z) =
∞
⊕
13.9.4 H i (X, Z),
i=0
a graded ring. There are also homology groups H i (X, Z) of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a k -multilinear form and an l multilinear form to get a ( k + l )-multilinear form. The ring structure in cohomology provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ringtheoretic properties. Birational geometry studies maps between the subrings of the function field. 13.9.5 Face ring of a simplicial complex
Every simplicial complex hasan associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize thenumbers of faces in each dimension of simplicial polytopes.
13.10 13.9.2 Burnside ring of a group
To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring’s additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
Function field of an irreducible algebraic variety
Category theoretical description
Main article: Category of rings Every ring can be thought of as a monoid in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product of Z -modules). The monoid action of a ring R on an abelian group is simply an R -module. Essentially, an R -module is a generalization of the notion of a vector space – where rather than a vector space over a field, one has a “vector space over a ring”. Let (A, +) be an abelian group and let End( A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f · g:
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CHAPTER 13. RING
• ( f + g)( x ) = f ( x ) + g( x )
13.11.3
• ( f · g)( x ) = f ( g( x ))
A semiring is obtained by weakening the assumption that (R,+) is an abelian group to the assumption that ( R,+) is a commutative monoid, and adding the axiom that 0 · a = a · 0 = 0 for all a in R (since it no longer follows from the other axioms). Example: a tropical semiring.
where + as in f ( x ) + g( x ) is addition in A, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversely, given any ring, ( R, +, · ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of ( R, +), by right (or left) distributivity. Let A = (R, +). Consider those endomorphisms of A, that “factor through” right (or left) multiplication of R . In other words, let EndR(A) be the set of all morphisms m of A, having the property that m(r · x ) = r · m( x ). It was seen that every r in R gives rise to a morphism of A: right multiplication by r . It is in fact true that this association of any element of R, to a morphism of A, as a function from R to EndR(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X -group (by X -group, it is meant a group with X being its set of operators).[45] In essence, the most general form of a ring, is the endomorphism group of some abelian X -group. Any ring can be seen as a preadditive category with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms.
13.11
Generalization
13.12
Semiring
Other ring-like objects
13.12.1 Ring object in a category
Let C be a category with finite products. Let pt denote a terminal object of C (an empty product). A ring object a in C is an object R equipped with morphisms R × R→ R 0 m (addition), R × R→R (multiplication), pt →R (additive 1 i R (additive inverse), and pt →R (multiidentity), R→ plicative identity) satisfyingthe usual ring axioms. Equivalently, a ring object is an object R equipped with a factorization of its functor of points hR = Hom(−, R) : C op → Sets through the category of rings: C op → forgetful Rings −→ Sets . 13.12.2 Ring scheme
In algebraic geometry, a ring scheme over a base scheme S is a ring object in the category of S -schemes. One example is the ring scheme W n over Spec Z, which for any commutative ring A returns the ring Wn(A) of p-isotypic Witt vectors of length n over A.[47] 13.12.3 Ring spectrum
In algebraic topology, a ring spectrum is a spectrum X toAlgebraists have defined structures more general than gether with a multiplication µ : X ∧ X → X and a unit rings by weakening or dropping some of ring axioms. map S → X from the sphere spectrum S , such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category 13.11.1 Rng of symmetric spectra. A rng is the same as a ring, except that the existence of a multiplicative identity is not assumed.[46]
13.13
13.11.2
Nonassociative ring
A nonassociative ring is an algebraic structure that satisfies all of the ring axioms but the associativity and the existence of a multiplicative identity. A notable example is a Lie algebra. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.
See also
• Algebra over a commutative ring • Algebraic structure • Categorical ring • Category of rings • Glossary of ring theory
83
13.15. CITATIONS
• Nonassociative ring • Ring theory • Semiring • Spectrum of a ring • Simplicial commutative ring Special types of rings:
• Boolean ring • Commutative ring • Dedekind ring • Differential ring • Division ring (skew field) • Exponential ring • Field • Integral domain • Lie ring • Local ring • Noetherian and artinian rings • Ordered ring • Principal ideal domain (PID) • Reduced ring • Regular ring • Ring of periods • Ring theory • SBI ring • Unique factorization domain (UFD) • Valuation ring and discrete valuation ring • Zero ring 13.14
Notes
^ a: Some authors only require that a ring be a semigroup
under multiplication; that is, do not require that there be a multiplicative identity (1). See the section Notes on the definition for more details. ^ b: Elements which do have multiplicative inverses are called units, see Lang 2002, §II.1, p. 84. ^ c: The closure axiom is already implied by the condition that +/• be a binary operation. Some authors therefore omit this axiom. Lang 2002
The transition from the integers to the rationals by adding fractions is generalized by the quotient field. ^ e: Many authors include commutativity of rings in the set of ring axioms (see above) and therefore refer to “commutative rings” as just “rings”. ^ d:
13.15
Citations
[1] Implicit in the assumption that "+" is a binary operation is that 1) a + b is defined for all ordered pairs (a,b) of elements a and b of R; 2) "+" is well-defined, that is, if a + b = c 1 and a + b = c 2 , then c 1 = c 2 ; and 3) R is closed under "+", meaning that for any a and b in R, the value of a + b is defined to be an element of R. The same applies to multiplication. Closure would be an axiom, however, only if, instead of binary operations on R, we had functions "+" and "·" a priori taking values in some larger set S . [2] Nicolas Bourbaki (1970). "§I.8”. Algebra. SpringerVerlag. [3] Saunders MacLane; Garrett Birkhoff (1967). Algebra. AMS Chelsea. p. 85. [4] Serge Lang (2002). Algebra (Third ed.). Springer-Verlag. p. 83. [5] The existence of 1 is not assumed by some authors. In this article, and more generally in Wikipedia, we adopt the most common convention of the existence of a multiplicative identity, and use the term rng if this existence is not required. See next subsection [6] I. M. Isaacs, Algebra: A Graduate Course, AMS, 1994, p. 160. [7] The development of Ring Theory [8] Kleiner 1998, p. 27. [9] Hilbert 1897. [10] [11] Cohn, Harvey (1980), Advanced Number Theory, New York: Dover Publications, p. 49, ISBN 978-0-48664023-5 [12] Fraenkel, pp. 143–145 [13] Jacobson (2009), p. 86, footnote 1. [14] Fraenkel, p. 144, axiom R₈₎. [15] Noether, p. 29. [16] Fraenkel, p. 144, axiom R₇₎. [17] Van der Waerden, 1930. [18] Zariski and Samuel, 1958. [19] Artin, p. 346. [20] Atiyah and MacDonald, p. 1.
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[21] Bourbaki, p. 96. [22] Eisenbud, p. 11. [23] Lang, p. 83. [24] Gardner and Wiegandt 2003. [25] Wilder 1965, p. 176. [26] Rotman 1998, p. 7. [27] This is the definition of Bourbaki. Some other authors such as Lang require a zero divisor to be nonzero. [28] In the unital case, like addition and multiplication, the multiplicative identity must be restricted from the original ring. The definition is also equivalent to requiring the set-theoretic inclusion is a ring homomorphism. [29] Cohn 2003, Theorem 4.5.1 [30] such a central idempotent is called centrally primitive. [31] Jacobson 1974, Theorem 2.10 [32] Bourbaki Algèbre commutative, Ch 5. §1, Lemma 2 [33] Cohn 2003, 4.4 [34] Lang 2002, Ch. XVII. Proposition 1.1. [35] Cohn 1995, Proposition 1.3.1.
• Bourbaki, N. (1998). Springer.
Algebra I, Chapters 1-3.
• Cohn, Paul Moritz (2003), Basic algebra: groups, rings, and fields , Springer, ISBN 978-1-85233-587-
8.
• Eisenbud, David (1995). Commutative algebra with a view toward algebraic geometry . Springer.
• Gardner, J.W.; Wiegandt, R. (2003). Radical Theory of Rings. Chapman & Hall/CRC Pure and Ap-
plied Mathematics. ISBN 0824750330.
• Herstein, I. N. (1994) [reprint of the 1968 original]. Noncommutative rings. Carus Mathematical Mono-
graphs 15 . With an afterword by Lance W. Small. Mathematical Association of America. ISBN 088385-015-X.
• Jacobson, Nathan (2009). Basic algebra 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1.
• Jacobson,
Nathan (1964). “Structure of rings”.
American Mathematical Society Colloquium Publications (Revised ed.) 37 .
• Jacobson, Nathan (1943). “The Theory of Rings”.
[36] Eisenbud 2004, Exercise 2.2
American Mathematical Society Mathematical Surveys I.
[37] Milne 2012, Proposition 6.4
• Kaplansky, Irving (1974), Commutative rings (Re-
[38] Milne 2012, The end of Chapter 7 [39] Atiyahand Macdonald, Theorem 10.17and its corollaries. [40] Cohn 1995, pg. 242. [41] Lang 2002, Ch XIV, §2 [42] Weibel, Ch 1, Theorem 3.8 [43] Milne CFT, Ch IV, §2 [44] Serre, J-P ., Applications algébriques de la cohomologie des groupes, I, II, Séminaire Henri Cartan, 1950/51 [45] Jacobson (2009), p. 162, Theorem 3.2. [46] Jacobson 2009. [47] Serre, p. 44.
13.16 13.16.1
References General references
• Artin, Michael (1991). Algebra. Prentice-Hall. • Atiyah, Michael; Macdonald, Ian G. (1969). Introduction to commutative algebra. Addison–Wesley.
vised ed.), University of Chicago Press, ISBN 0226-42454-5, MR 0345945.
• Lam, Tsit Yuen (2001). A first course in noncom-
mutative rings. Graduate Texts in Mathematics 131
(2nd ed.). Springer. ISBN 0-387-95183-0.
• Lam, Tsit Yuen (2003). Exercises in classical ring
theory. Problem Books in Mathematics (2nd ed.).
Springer. ISBN 0-387-00500-5.
• Lam,
Tsit Yuen (1999). Lectures on modules and rings. Graduate Texts in Mathematics 189. Springer. ISBN 0-387-98428-3.
• Lang, Serge (2002), Algebra, Graduate Texts in
Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, Zbl 0984.00001, MR 1878556.
• Matsumura, Hideyuki (1989). Commutative Ring
Theory. Cambridge Studies in Advanced Math-
ematics (2nd ed.). Cambridge University Press. ISBN 978-0-521-36764-6.
• Milne, J. “A primer of commutative algebra”. • Rotman, Joseph (1998), Galois Theory (2nd ed.), Springer, ISBN 0-387-98541-7.
85
13.16. REFERENCES
• van der Waerden, Bartel Leendert (1930), Moderne Algebra. Teil I , Die Grundlehren der mathema-
tischen Wissenschaften 33, Springer, ISBN 9783-540-56799-8, MR 0009016MR 0037277MR 0069787MR 0122834MR 0177027MR 0263581.
• Wilder, Raymond Louis (1965). Introduction to Foundations of Mathematics . Wiley.
• Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra 1. Van Nostrand.
• Milne, J. “Class field theory”. • Nagata, Masayoshi (1962) [1975 reprint], Local rings, InterscienceTracts in Pure andApplied Math-
ematics 13, Interscience Publishers, ISBN 978-088275-228-0, MR 0155856.
• Pierce, Richard S. (1982). Associative algebras . Graduate Texts in Mathematics 88. Springer. ISBN 0-387-90693-2.
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics 67, Springer.
13.16.2
Special references
• Balcerzyk,
Stanisław; Józefiak, Tadeusz (1989), Commutative Noetherian and Krull rings, Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7.
• Balcerzyk,
Stanisław; Józefiak, Tadeusz (1989), Dimension, multiplicity and homological methods , Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2.
• Ballieu, R. (1947). “Anneaux finis; systèmes hyper-
complexes de rang trois sur un corps commutatif”. Ann. Soc. Sci. Bruxelles I (61): 222–227.
• Berrick, A. J.; Keating, M. E. (2000). An Introduc-
tion to Rings and Modules with K-Theory in View .
Cambridge University Press.
• Cohn, Paul Moritz (1995), Skew Fields: Theory of General Division Rings , Encyclopedia of Mathemat-
ics and its Applications 57, Cambridge University Press, ISBN 9780521432177.
• Springer, Tonny A. (1977), Invariant theory, Lecture Notes in Mathematics 585, Springer.
• Weibel, Charles. “The K-book: An introduction to algebraic K-theory”.
• Zariski, Oscar; Samuel, Pierre (1975). Commuta-
tive algebra. Graduate Texts in Mathematics. 28-
29. Springer. ISBN 0-387-90089-6. 13.16.3 Primary sources
• Fraenkel, A. (1914). "Über die Teiler der Null und die Zerlegung von Ringen”. J. reine angew. Math. 145: 139–176.
• Hilbert,
David (1897). “Die Theorie der algebraischen Zahlkörper”. Jahresbericht der Deutschen Mathematiker Vereinigung 4.
• Noether,
Emmy (1921). “Idealtheorie in Ringbereichen”. Math. Annalen 83: 24–66. doi:10.1007/bf01464225.
• Eisenbud,
David (1995), Commutative algebra. With a view toward algebraic geometry. , Graduate 13.16.4 Historical references Texts in Mathematics 150 , Springer, ISBN 978-0387-94268-1, MR 1322960. • History of ring theory at the MacTutor Archive
• Gilmer, R.; Mott, J. (1973).
“Associative Rings of Order”. Proc. Japan Acad. 49: 795–799. doi:10.3792/pja/1195519146 .
• Harris, J. W.; Stocker, H. (1998). Handbook of Mathematics and Computational Science . Springer.
• Jacobson, Nathan (1945), “Structure theory of alge-
braic algebras of bounded degree”, Annals of Mathematics (Annals of Mathematics) 46 (4): 695–707, doi:10.2307/1969205, ISSN 0003-486X, JSTOR 1969205.
• Knuth,D.E. (1998). The Art of Computer Program-
• Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, 1996.
• Bronshtein,
I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004. ISBN 3-540-43491-7.
• Faith, Carl, Rings and things and a fine array of twentieth century associative algebra . Mathematical
Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0-8218-0993-8.
ming. Vol. 2: Seminumerical Algorithms (3rd ed.).
• Itô, K. (Ed.). “Rings.” §368 in Encyclopedic Dictio-
• Korn, G. A.; Korn, T. M. (2000). Mathematical
• Kleiner, I., “The Genesis of the Abstract Ring Con-
Addison–Wesley.
Handbook for Scientists and Engineers . Dover.
nary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986. cept”, Amer. Math. Monthly 103, 417–424, 1996.
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• Kleiner, I., “From numbers to rings: the early his-
tory of ring theory”, Elem. Math. 53 (1998), 18–35.
• Renteln, P. and Dundes, A. “Foolproof:
A Sampling of Mathematical Folk Humor.” Notices Amer. Math. Soc. 52, 24–34, 2005.
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D. and Bloom, D. M. “Problem E1648.” Amer. Math. Monthly 71, 918–920, 1964.
• Van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985.
Chapter 14
Subgroup This article is about the mathematical concept. For the galaxy-related concept, see galaxy group.
• The above condition can be stated in terms of a
In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H . This is usually denoted H ≤ G , read as "H is a subgroup of G ". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (i.e. H ≠ G ). This is usually represented notationally by H < G , read as "H is a proper subgroup of G ". Some authors also exclude the trivial group from being proper (i.e. {e} ≠ H ≠ G ).[1][2] If H is a subgroup of G , then G is sometimes called an overgroup of H . The same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the ordered pair ( G , ∗), usually to emphasize the operation ∗ when G carries multiple algebraic or other structures. This article will write ab for a ∗ b, as is usual.
14.1
Basic properties of subgroups
homomorphism; that is, H is a subgroup of a group G if and only if H is a subset of G and there is an inclusion homomorphism (i.e., i( a) = a for every a) from H to G .
• The identity of a subgroup is the identity of the
group: if G is a group with identity eG , and H is a subgroup of G with identity eH , then eH = eG .
• The inverse of an element in a subgroup is the in-
verse of the element in the group: if H is a subgroup ofagroup G , and a and b areelements of H such that ab = ba = eH , then ab = ba = eG .
• The intersection of subgroups A and B is again a [3]
subgroup. The union of subgroups A and B is a subgroup if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z but their sum 5 is not. Another example is the union of the x-axis and the y-axis in the plane (with the addition operation); each of these objects is a subgroup but their union is not. This also serves as an example of two subgroups, whose intersection is precisely the identity.
• If S is a subset of G , then there exists a minimum
subgroup containing S , whichcanbefoundbytaking the intersection of all of subgroups containing S ; it is denoted by and is said to be the subgroup generated by S . An element of G is in if and onlyif it is a finite product of elements of S and their inverses.
• Every element a of a group G generates the cyclic
• A subset H of the group G is a subgroup of G if and
only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H , then ab and a−1 are also in H . These two conditions can be combined into one equivalent condition: whenever a and b are in H , then ab −1 is also in H .) In the case that H is finite, then H is a subgroup if and only if H is closed under products. (In this case, every element a of H generates a finite cyclic subgroup of H , and the inverse of a is then a −1 = an − 1, where n is the order of a.) 87
subgroup . If is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which an = e,and n is called the order of a. If is isomorphic to Z,then a issaidtohave infinite order .
• The subgroups of any given group form a complete
lattice under inclusion, called the lattice of subgroups. (While the infimum here is the usual settheoretic intersection, the supremum of a set of subgroups is the subgroup generated by the set-theoretic union of the subgroups, not the set-theoretic union
88
CHAPTER 14. SUBGROUP
itself.) If e is the identity of G , then the trivial Right cosets are defined analogously: Ha = {ha : h in group {e} is the minimum subgroup of G , while the H }. They are also the equivalence classes for a suitable maximum subgroup is the group G itself. equivalence relation and their number is equal to [ G : H ]. If aH = Ha for every a in G , then H is said to be a normal subgroup. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal.
G
H
0
4
1
5 1+H
2
6 2+H
3
7 3+H
G is the group Z /8Z , the integers mod 8 under addition. The subgroup H contains only 0 and 4, and is isomorphic to Z /2Z . There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (written using additive notation since this is an additive group). Together they partition the entire group G into equal-size, nonoverlapping sets. The index [G : H] is 4.
14.2
14.3 Example: Subgroups of Z8 Let G be the cyclic group Z8 whose elements are G = 0, 2, 4, 6, 1, 3, 5, 7
{
}
and whose group operation is addition modulo eight. Its Cayley table is This group has two nontrivial subgroups: J ={0,4} and H ={0,2,4,6}, where J is also a subgroupof H . The Cayley table for H is the top-left quadrant of the Cayley table for G . The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
14.4 Example: Subgroups of S₄ (the symmetric group on 4 elements)
Every group has as many small subgroups as neutral elements on the main diagonal: Cosets and Lagrange’s theoThe trivial group and two-element groups Z2 . These rem small subgroups are not counted in the following list.
Given a subgroup H and some a in G, we define the left coset aH = {ah : h in H }. Because a is invertible, themap φ : H → aH given by φ(h) = ah is a bijection. Furthermore, every element of G is contained in precisely one left coset of H ; the left cosets are the equivalence classes corresponding to the equivalence relation a 1 ~ a 2 if and only if a1−1 a2 is in H . The number of left cosets of H is called the index of H in G and is denoted by [ G : H ]. Lagrange’s theorem states that for a finite group G and a subgroup H ,
| G| [G : H ] = |H | where |G | and |H | denote the orders of G and H , respectively. In particular, the order of every subgroup of G (and the order of every element of G ) must be a divisor of |G |.
14.4.1
12 elements
14.4.2
8 elements
14.4.3
6 elements
14.4.4
4 elements
14.4.5
3 elements
14.5
Other examples
• An ideal in a ring R is a subgroup of the additive group of R .
• Let A be an abelian group; the elements of A that have finite period form a subgroup of A called the torsion subgroup of A .
14.8. REFERENCES
The alternating group A4 showing only the even permutations Subgroups:
14.6 See also
• Cartan subgroup • Fitting subgroup • Stable subgroup • Fixed-point subgroup 14.7
Notes
[1] Hungerford (1974), p. 32 [2] Artin (2011), p. 43 [3] Jacobson (2009), p. 41
14.8 References
• Jacobson, Nathan (2009), Basic algebra 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1.
• Hungerford,
Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181.
• Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770.
89
Chapter 15
Symmetry For other uses, see Symmetry (disambiguation). Symmetry (from Greek συμμετρία symmetria “agree-
SYMMETRIC
ASYMMETRIC
4
3
2
3 Leonardo da Vinci 's ' VitruvianMan' (ca. 1487) isoften usedas a representationof symmetryin the humanbody and, by extension, the natural universe.
O
432
Sphere symmetrical group o representing an octahedral rotational symmetry. The yellow region shows the fundamental domain.
ment in dimensions, due proportion, arrangement”)[1] in everyday language refers to a sense of harmonious and beautiful proportion and balance.[2][lower-alpha 1] In mathematics, “symmetry” has a more precise definition, that an object is invariant to a transformation, such as reflection but including other transforms too. Although these two meanings of “symmetry” can sometimes be told apart, they are related, so they are here discussed together.
Mathematical symmetry may be observed with respect to the passage of time; a s a spatial relationship; through geometric transformations such as scaling, reflection, and rotation; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.[3][lower-alpha 2] This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music. The opposite of symmetry is asymmetry.
90
91
15.1. IN MATHEMATICS
15.1.1
In geometry
Main article: Symmetry (geometry) A geometric shape or object is symmetric if it can be di-
A fractal -like shape that has reflectional symmetry , rotational symmetry and self-similarity , three forms of symmetry. This shape is obtained by a finite subdivision rule. The triskelion has 3-fold rotational symmetry.
vided into two or more identical pieces that are arranged in an organized fashion. [4] This means that an object is symmetric if there is a transformation that moves individual pieces of the object but doesn't change the overall shape. The type of symmetry is determined by the way thepieces areorganized, or by thetype of transformation:
• An object has reflectional symmetry (line or mirror symmetry) if there is a line going through it which divides it into two pieces which are mirror images of each other.[5]
• An object has rotational symmetry if the object can be rotated about a fixed point without changing the overall shape.[6]
• An object has translational symmetry if it can be [7] translated without changing its overall shape.
• An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.[8]
• An object has scale symmetry if it does not change [9] shape when it is expanded or contracted. Fractals also exhibit a form of scale symmetry, where small portions of the fractal are similar in shape to large portions.[10]
Symmetric arcades of a portico in the Great Mosque of Kairouan also called the Mosque of Uqba, in Tunisia.
• Other symmetries include glide reflection symmetry and rotoreflection symmetry.
15.1.2
15.1
In mathematics
In logic
A dyadic relation R is symmetric if and only if, whenever it’s true that Rab, it’s true that Rba.[11] Thus, “is the same
92
CHAPTER 15. SYMMETRY
ageas”issymmetrical,forifPaulisthesameageasMary, symmetries of particles; and supersymmetry of physical then Mary is the same age as Paul. theories. Symmetric binary logical connectives are and (∧, or &), or (∨, or |), biconditional (if and only if) (↔), nand (notand, or ⊼), xor (not-biconditional, or ⊻), and nor (not-or, or ⊽). 15.1.3 Other areas of mathematics
Main article: Symmetry (mathematics) Generalizing from geometrical symmetry in the previous section, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object.[12] The set of operations that preserve a given property of the object form a group. In general, every kind of structure in mathematics will have its own kind of symmetry. Examples include even and odd functions in calculus; the symmetric group in abstract algebra; symmetric matrices in linear algebra; and the Galois group in Galois theory. In statistics, it appears as symmetric probability distributions, and as skewness, asymmetry of distributions.
15.2 In science and nature Further information: Patterns in nature
15.2.1
Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.
In physics
Main article: Symmetry in physics
15.2.2
In biology
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations.[13] This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate PW Anderson to write in his widely read 1972 article More is Different that “it is only slightly overstating the case to say that physics is the study of symmetry.” [14] See Noether’s theorem (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; a conserved current, in Noether’s original language);[15] and also, Wigner’s classification, which says that the symmetries of the laws of physics determine the properties of the particles found in nature.[16] Important symmetries in physics include continuous symmetries and discrete symmetries of spacetime; internal
Further information: symmetry in biology and facial symmetry Bilateral animals, including humans, are more or less symmetric with respect to the sagittal plane which divides the body into left and right halves. [17] Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The headbecomesspecialized withamouthandsenseorgans, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.[18] Plants and sessile (attached) animals such as sea anemones often have radial or rotational symmetry, which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the echinoderms, thegroup that includes starfish, sea urchins, and sea lilies.[19]
93
15.4. IN THE ARTS
15.2.3
In chemistry
Further information: Mathematics and art
Main article: molecular symmetry 15.4.1 In architecture Symmetry is important to chemistry because it undergirds essentially all specific interactions between Further information: Mathematics and architecture molecules in nature (i.e., via the interaction of natural Symmetry finds its ways into architecture at every scale, and human-made chiral molecules with inherently chiral biological systems). The control of the symmetry of molecules produced in modern chemical synthesis contributes to the ability of scientists to offer therapeutic interventions with minimal side effects. A rigorous understanding of symmetry explains fundamental observations in quantum chemistry, and in the applied areas of spectroscopy and crystallography. The theory and application of symmetry to these areas of physical science draws heavily on the mathematical area of group theory.[20]
15.3 In social interactions People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of Reciprocity, empathy, sympathy, apology, dialog, respect, justice, and revenge. Reflectiveequilibrium isthebalancethatmaybe attained through deliberative mutual adjustment among general principles and specific judgments.[21] Symmetrical interactions send the moral message “we are all the same” while asymmetrical interactions maysend the message “I am special; better than you.” Peer relationships, such as can be governed by the golden rule, are based on symmetry, whereas power relationships are based on asymmetry. [22] Symmetrical relationships can to some degree be maintained by simple (game theory) strategies seen in symmetric games such as tit for tat.[23]
Seen from the side, the Taj Mahal has bilateral symmetry; from the top (in plan), it has fourfold symmetry.
from the overall external views of buildings such as Gothic cathedrals and The White House, through the layout of theindividualfloor plans,anddowntothedesignof individual building elements such as tile mosaics. Islamic buildings such as the Taj Mahal and the Lotfollah mosque make elaborate use of symmetry both in their structure and in their ornamentation.[24][25] Moorish buildings like the Alhambra are ornamented with complex patterns made using translational and reflection symmetries as well as rotations. [26] It has been said that only bad architects rely on a “symmetrical layout of blocks, masses and structures"; [27] Modernist architecture, starting with International style, relies instead on “wings and balance of masses”. [27] 15.4.2 In pottery and metal vessels
15.4
In the arts
Claypotsthrown on a pottery wheel acquire rotational symmetry. The ceiling of Lotfollah mosque , Isfahan , Iran has 8-fold symmetries.
Since theearliest uses of pottery wheels tohelpshapeclay
94
CHAPTER 15. SYMMETRY
vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives. Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient Chinese, for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.[28] 15.4.3
In quilts
dians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugstypically use quadrilateral symmetry—that is, motifs that are reflected across both the horizontal and vertical axes.[30][31] 15.4.5
In music
Major and minor triads on the white piano keys are symmetrical to the D. (compare article) (file) Symmetry is not restricted to the visual arts. Its role in the history of music touches many aspects of the creation and perception of music. Musical form
Symmetry has been used as a formal constraint by many composers, such as the arch (swell) form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney. In classical music, Bach used the symmetry concepts of permutation and invariance.[32] Pitch structures
Kitchen Kaleidoscope Block
As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. [29] 15.4.4
In carpets and rugs
Persian rug with quadrilateral symmetry
A long tradition of the use of symmetry in carpet and rug patterns spans a variety of cultures. American Navajo In-
Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. However, composers such as Alban Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal centers. Perle (1992)[33] explains “C–E, D–F♯, [and] Eb–G, are different instances of the same interval … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:" Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0). Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin, Edgard Varèse, and the
95
15.5. SEE ALSO
Vienna school. At the same time, these progressions sig- 15.5 See also nal the end of tonality. • Burnside’s lemma The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg’s • Chirality Quartet , Op. 3 (1910).[34] • Even and odd functions Equivalency
Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm. 15.4.6
In other arts and crafts
• Fixed points of isometry groups in Euclidean space – center of symmetry
• Isotropy • Spacetime symmetries • Spontaneous symmetry breaking • Symmetry-breaking constraints • Symmetric relation • Symmetries of polyiamonds • Symmetries of polyominoes • Symmetry group • Time symmetry • Wallpaper group
Celtic knotwork
Symmetries appear in the design of objects of all kinds. 15.6 Notes Examples include beadwork, furniture, sand paintings, knotwork, masks, and musical instruments. Symmetries [1] For example, Aristotle ascribed spherical shape to the heavenly bodies, attributing this formally defined geometare central to the art of M.C. Escher and the many apric measure of symmetry to the natural order and perfecplications of tessellation in art and craft forms such as tion of the cosmos. wallpaper, ceramic tilework, batik, ikat, carpet-making, and many kinds of textile and embroidery patterns.[35] [2] Symmetric objects can be material, such as a person, 15.4.7
In aesthetics
crystal, quilt, floor tiles, or molecule, or it can be an abstract structure such as a mathematical equation or a series of tones (music).
Main article: Symmetry (physical attractiveness)
15.7 References The relationship of symmetry to aesthetics is complex. Humans find bilateral symmetry in faces physically attractive;[36] it indicates health and genetic fitness. [37][38] Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. People prefer shapes that have some symmetry, but enough complexity to make them interesting.[39] 15.4.8
In literature
Symmetry can be found in various forms in literature, a simple example being the palindrome where a brief text reads the same forwards or backwards. Stories may have a symmetrical structure, as in the rise:fall pattern of Beowulf .
[1] “symmetry”. Online Etymology Dictionary. [2] Zee, A. (2007). Fearful Symmetry. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-13482-6. [3] Mainzer, Klaus (2005). Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science. World Scientific. ISBN 981-256-192-7. [4] E. H. Lockwood, R. H. Macmillan, Geometric Symmetry, London: Cambridge Press, 1978 [5] Weyl, Hermann (1982) [1952]. Symmetry. Princeton: Princeton University Press. ISBN 0-691-02374-3. [6] Singer, David A. (1998). Geometry: Plane and Fancy. Springer Science & Business Media.
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[7] Stenger, Victor J. (2000) and Mahou Shiro (2007). Time- [26] Derry, Gregory N. (2002). What Science Is and How It less Reality. Prometheus Books. Especially chapter 12. Works. Princeton University Press. pp. 269–. ISBN 978Nontechnical. 1-4008-2311-6. [8] Bottema, O, and B. Roth, Theoretical Kinematics, Dover [27] Dunlap, David W. (31 July 2009). “Behind the Scenes: Publications (September 1990) Edgar Martins Speaks”. New York Times. Retrieved 11 November 2014. “My starting point for this construction [9] Tian Yu Cao Conceptual Foundations of Quantum Field was a simple statement which I once read (and which does Theory Cambridge University Press p.154-155 not necessarily reflect my personal views): ‘Only a bad architect relies on symmetry; instead of symmetrical layout [10] Gouyet, Jean-François (1996). Physics and fractal strucof blocks, masses and structures, Modernist architecture tures. Paris/New York: Masson Springer. ISBN 978-0relies on wings and balance of masses.’ 387-94153-0. [11] Josiah Royce, Ignas K. Skrupskelis (2005) The Basic [28] The Art of Chinese Bronzes. Chinavoc (2007-11-19). Retrieved on 2013-04-16. Writings of Josiah Royce: Logic, loyalty, and community (Google eBook) Fordham Univ Press, p. 790 [29] Quate: Exploring Geometry Through Quilts. [12] Christopher G. Morris (1992) Academic Press Dictionary Its.guilford.k12.nc.us. Retrieved on 2013-04-16. of Science and Technology Gulf Professional Publishing [30] Marla Mallett Textiles & Tribal Oriental Rugs. The [13] Costa, Giovanni; Fogli, Gianluigi (2012). Symmetries Metropolitan Museum of Art, New York. and Group Theory in Particle Physics: An Introduction to Space-Time and Internal Symmetries. Springer Science & [31] Dilucchio: Navajo Rugs. Navajocentral.org (2003-10-
Business Media. p. 112.
26). Retrieved on 2013-04-16.
[14] Anderson, P.W. (1972). “More is Dif- [32] see (“Fugue No. 21,” pdf or Shockwave) ferent” (PDF). Science 177 (4047): 393– 396. Bibcode:1972Sci...177..393A. [33] Perle, George (1992). “Symmetry, the twelve-tone scale, doi:10.1126/science.177.4047.393. PMID 17796623. and tonality”. Contemporary Music Review 6 (2): 81–96. doi:10.1080/07494469200640151. [15] Kosmann-Schwarzbach, Yvette (2010). The Noether theorems: Invariance and conservation laws in the twentieth [34] Perle, George (1990). The Listening Composer . Univercentury. Sources and Studies in the History of Mathematsity of California Press. ics and Physical Sciences. Springer-Verlag. ISBN 978-0387-87867-6. [35] Cucker, Felix (2013). Manifold Mirrors: The Crossing Paths of the Arts and Mathematics. Cambridge University [16] Wigner, E. P. (1939), “On unitary representations of Press. pp. 77–78, 83, 89, 103. ISBN 978-0-521-72876the inhomogeneous Lorentz group”, Annals of Mathemat8. ics 40 (1): 149–204, Bibcode:1939AnMat..40..149W, doi:10.2307/1968551, MR 1503456. [36] Grammer, K.; Thornhill, R. (1994). “Human (Homo sapiens) facial attractiveness and sexual selection: the role [17] Valentine, James W. “Bilateria”. AccessScience. Reof symmetry and averageness”. Journal of Comparative trieved 29 May 2013. Psychology (Washington, D.C.) 108 (3): 233–42. [18] Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan (2002). “Animal Diversity (Third Edition)" (PDF). [37] Rhodes, Gillian; Zebrowitz, Leslie, A. (2002). Facial Attractiveness - Evolutionary, Cognitive, and Social PerspecChapter 8: Acoelomate Bilateral Animals . McGraw-Hill. tives. Ablex. ISBN 1-56750-636-4. p. 139. Retrieved October 25, 2012. [19] Stewart, Ian (2001). What Shape is a Snowflake? Magical [38] Jones, B. C., Little, A. C., Tiddeman, B. P., Burt, D. M., & Perrett, D. I. (2001). Facial symmetry and judgements Numbers in Nature. Weidenfeld & Nicolson. pp. 64–65. of apparent health Support for a “‘ good genes ’” expla[20] Lowe, John P; Peterson, Kirk (2005). Quantum Chemistry nation of the attractiveness – symmetry relationship, 22, (Third ed.). Academic Press. ISBN 0-12-457551-X. 417–429. [21] Daniels, Norman (2003-04-28). “Reflective Equilib- [39] Arnheim, Rudolf (1969). Visual Thinking. University of rium”. Stanford Encyclopedia of Philosophy. California Press. [22] Emotional Competency: Symmetry [23] Lutus, P. (2008). “The Symmetry Principle”. Retrieved 28 September 2015. [24] Williams: Symmetry in Architecture. Members.tripod.com(1998-12-31). Retrieved on 2013-04-16. [25] Aslaksen: Mathematics in Art and Architecture. Math.nus.edu.sg. Retrieved on 2013-04-16.
15.8
Further reading
• The Equation That Couldn't Be Solved: How Mathe-
matical Genius Discovered the Language of Symmetry, Mario Livio, Souvenir Press 2006, ISBN 0-285-
63743-6
15.9. EXTERNAL LINKS
15.9
External links
• Dutch: Symmetry Around a Point in the Plane • Chapman: Aesthetics of Symmetry • ISIS Symmetry
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Chapter 16
Symmetry group 16.1 Introduction Not to be confused with Symmetric group. This article is about the abstract algebraic structures. For other uses, see Symmetry group (disambiguation). In abstract algebra, the symmetry group of an object The “objects” may be geometric figures, images, and patterns, such as a wallpaper pattern. The definition can be made more precise by specifying what is meant by image or pattern, e.g., a function of position with values in a set of colors. For symmetry of physical objects, one may also want to take their physical composition into account. The group of isometries of space induces a group action on objects in it. The symmetry group is sometimes also called full symmetry group in order to emphasize that it includes the orientation-reversing isometries (like reflections, glide reflections and improper rotations)underwhichthefigureis invariant. The subgroup of orientation-preserving isometries (i.e. translations, rotations, and compositions of these) that leave the figure invariant is called its proper symmetry group . The proper symmetry group of an object is equal to its full symmetry group if and only if the object is chiral (and thus there are no orientationreversing isometries under which it is invariant). Any symmetry group whose elements have a common fixed point, which is true for all finite symmetry groups and also for the symmetry groups of bounded figures, can be represented as a subgroup of the orthogonal group O(n) by choosing the origin to be a fixed point. The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is therefore also called rotation group of the figure. A tetrahedron is invariant under 12 distinct rotations , reflections A discrete symmetry group is a symmetry group such excluded. These are illustrated here in the cycle graph format, that for every point of the space the set of images of the along with the 180° edge (blue arrows) and 120° vertex (reddish point under the isometries in the symmetry group is a arrows) rotations that permute the tetrahedron through the posi- discrete set. tions. The 12 rotations form the rotation (symmetry) group of Discrete symmetry groups come in three types: (1) fithe figure. nite point groups, which include only rotations, reflections, inversion and rotoinversion – they are just the fi(image, signal, etc.) is the group of all transformations nite subgroups of O(n), (2) infinite lattice groups, which under which the object is invariant with composition as include only translations, and (3) infinite space groups the group operation. For a space with a metric, it is a which combines elements of both previous types, and subgroup of the isometry group of the space concerned. may also include extra transformations like screw axis If not stated otherwise, this article considers symmetry and glide reflection. There are also continuous symmetry groups in Euclidean geometry, but the concept may also groups, which contain rotations of arbitrarily small angles be studied in more general contexts as expanded below. or translationsof arbitrarily small distances. The group of 180°
180°
180°
180°
180°
180°
120°
120°
120°
120°
120°
120°
120°
120°
120°
120°
120°
120°
98
99
16.3. TWO DIMENSIONS
all symmetries of a sphere O(3) is an example of this, and in general such continuous symmetry groups are studied as Lie groups. With a categorization of subgroups of the Euclidean group corresponds a categorization of symmetry groups. Two geometric figures are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of the Euclidean group E(n) (the isometry group of Rn ), where two subgroups H 1 , H 2 of a group G are conjugate, if there exists g ∈ G such that H 1 = g−1 H 2 g. For example:
• two 3D figures have mirror symmetry, but with respect to different mirror planes.
• two 3D figures have 3-fold rotational symmetry, but with respect to different axes.
• two 2D patterns have translational symmetry, each in one direction; the two translation vectors have the same length but a different direction.
When considering isometry groups, one may restrict oneself to those where for all points the set of images under the isometries is topologically closed. This includes all discrete isometry groups and also those involved in continuous symmetries, but excludes for example in 1D the group of translations by a rational number. A “figure” with this symmetry group is non-drawable and up to arbitrarily fine detail homogeneous, without being really homogeneous.
16.2
One dimension
The isometry groups in one dimension where for all points the set of images under the isometries is topologically closed are:
• the trivial group C1 • the groups of two elements generated by a reflection in a point; they are isomorphic with C 2
• the infinite discrete groups generated by a transla-
tion; they are isomorphic with Z, the additive group of the integers
• the infinite discrete groups generated by a transla-
tion and a reflection in a point; they are isomorphic with the generalized dihedral group of Z, Dih(Z), also denoted by D∞ (which is a semidirect product of Z and C2 ).
• the group generated by all translations (isomorphic
with the additive group of the real numbers R); this group cannot be the symmetry group of a “pattern": it would be homogeneous, hence could also be reflected. However, a uniform one-dimensional vector field has this symmetry group.
• the group generated by all translations and re-
flections in points; they are isomorphic with the generalized dihedral group of R, Dih(R).
See also symmetry groups in one dimension.
16.3 Two dimensions Up to conjugacy the discrete point groups in twodimensional space are the following classes:
• cyclic groups C1, C2, C3, C4, ... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/ n
• dihedral groups D1, D 2, D3, D4, ..., where Dn (of
order 2n) consists of the rotations in Cn together with reflections in n axes that pass through the fixed point.
C1 is the trivial group containing only the identity operation, which occurs when thefigurehasno symmetry at all, for example the letter F. C2 is the symmetry group of the letter Z, C3 that of a triskelion, C4 of a swastika, and C5 , C6 , etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four. D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter A. D2 , which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes. D3 , D4 etc. are the symmetry groups of the regular polygons. The actual symmetry groups in each of these cases have two degrees of freedom for the center of rotation, and in thecase of thedihedral groups, one more for thepositions of the mirrors. The remaining isometry groups in two dimensions with a fixed point, where for all points the set of images under the isometries is topologically closed are:
• the special orthogonal group SO(2) consisting of all
rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of Cn. There is no geometric figure that has as full symmetry group the circle group, but for a vector field it may apply (see thethree-dimensional case below).
100
CHAPTER 16. SYMMETRY GROUP
• the orthogonal group O(2) consisting of all rotations
16.5 Symmetry groups in general
aboutafixedpointandreflectionsinanyaxisthrough that fixed point. This is the symmetry group of a See also: Automorphism circle. It is also called Dih(S1 )asitisthe generalized dihedral group of S1. In wider contexts, a symmetry group maybeanykindof For non-bounded figures, the additional isometry groups transformation group, or automorphism group. Once weknowwhat kindof mathematical structure weareconcan include translations; the closed ones are: cerned with, weshould be able to pinpoint what mappings preserve the structure. Conversely, specifying the sym• the 7 frieze groups metry can define the structure, or at least clarify what • the 17 wallpaper groups we mean by an invariant, geometric language in which • for each of the symmetry groups in one dimension, to discuss it; this is one way of looking at the Erlangen the combination of all symmetries in that group in programme. one direction, and the group of all translations in the For example, automorphism groups of certain models of perpendicular direction finite geometries are not “symmetry groups” in the usual • ditto with also reflections in a line in the first direc- sense, although they preserve symmetry. They do this by preserving families of point-sets rather than point-sets (or tion “objects”) themselves. Like above, thegroup of automorphisms of space induces a group action on objects in it. 16.4 Three dimensions For a given geometric figure in a given geometric space, See also: Point groups in three dimensions consider the following equivalence relation: two automorphisms of space are equivalent if and only if the two Up to conjugacy the set of three-dimensional point images of the figure are the same (here “the same” does groups consists of 7 infinite series, and 7 separate ones. not mean something like e.g. “the same up to translation In crystallography they are restricted to be compatible and rotation”, but it means “exactly the same”). Then the with the discrete translation symmetries of a crystal lat- equivalence class of the identity is the symmetry group tice. This crystallographic restriction of the infinite fam- of the figure, and every equivalence class corresponds to ilies of general point groups results in 32 crystallographic one isomorphic version of the figure. point groups (27 from the 7 infinite series, and 5 of the 7 There is a bijection between every pair of equivalence others). classes: the inverse of a representative of the first equivThe continuous symmetry groups with a fixed point in- alence class, composed with a representative of the second. clude those of: In the case of a finite automorphism group of the whole • cylindrical symmetry without a symmetry plane per- space, its order is the order of the symmetry group of the pendicular to the axis, this applies for example often figure multiplied by the number of isomorphic versions for a bottle of the figure. • cylindrical symmetry with a symmetry plane per- Examples: pendicular to the axis
• spherical symmetry
• Isometries of the Euclidean plane, the figure is
a rectangle: there are infinitely many equivalence For objects and scalar fields the cylindrical symmetry imclasses; each contains 4 isometries. plies vertical planes of reflection. However, for vector fields it does not: in cylindrical coordinates with respect • The space is a cube with Euclidean metric; the figto some axis, A = Aρ ρ + A φ φ + A z z has cylindrical ures include cubes of thesame size as thespace, with symmetry with respect to the axis if and only if Aρ , Aφ , colors or patterns on the faces; the automorphisms and A z have this symmetry, i.e., they do not depend on of the space are the 48 isometries; the figure is a φ. Additionally there is reflectional symmetry if and only cube of which one face has a different color; the figif Aφ = 0 . ure has a symmetry group of 8 isometries, there are 6 equivalence classes of 8 isometries, for 6 isomorFor spherical symmetry there is no such distinction, it imphic versions of the figure. plies planes of reflection. The continuous symmetry groups without a fixed point include those with a screw axis, such as an infinite helix. Compare Lagrange’s theorem (group theory) and its See also subgroups of the Euclidean group. proof. ˆ
ˆ
ˆ
101
16.8. EXTERNAL LINKS
16.6 See also
• Crystallography • Crystallographic point group • Crystal system • Euclidean plane isometry • Fixed points of isometry groups in Euclidean space • Group action • Molecular symmetry • Permutation group • Point group • Space group • Symmetric group • Symmetry • Symmetry in quantum mechanics 16.7
Further reading
• Burns, G.; Glazer, A. M. (1990). Space Groups for Scientists and Engineers (2nd ed.). Boston: Aca-
demic Press, Inc. ISBN 0-12-145761-3.
• Clegg, W (1998). Crystal Structure Determination
(Oxford Chemistry Primer). Oxford: Oxford Uni-
versity Press. ISBN 0-19-855901-1.
• O'Keeffe, M.; Hyde, B. G. (1996). Crystal Struc-
tures; I. Patterns and Symmetry . Washington, DC: Mineralogical Society of America, Monograph Series. ISBN 0-939950-40-5.
• Miller, Willard Jr.
(1972). Symmetry Groups and Their Applications . New York: Academic Press. OCLC 589081. Retrieved 2009-09-28.
16.8
External links
• Weisstein,
Eric
• Weisstein,
Eric W., “Tetrahedral Group”,
MathWorld . MathWorld .
W., “Symmetry
Group”,
• Overview of the 32 crystallographic point groups -
form the first parts (apart from skipping n=5) of the 7 infinite series and 5 of the 7 separate 3D point groups
Chapter 17
Vector field tion of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.
17.1 A portion of the vector field (sin y , sin x)
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space.[1] A vector field in the plane (for instance), can be visualised as: a collection of arrows with a given magnitude and direction, each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. The elements of differential and integral calculus extend naturally to vector fields. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow). In coordinates, a vector field on a domain in ndimensional Euclidean space can be represented as a vector-valued function that associates an n -tuple of real numbers to each point of the domain. This representa-
Definition
17.1.1 Vector fields on subsets of Euclidean space
Two representations of the same vector field: v( x , y) = −r. The arrows depict the field at discrete points, however, the field exists everywhere. Given a subset S in R n , a vector field is represented by a vector-valued function V : S → R n in standard Cartesian coordinates ( x 1 , ..., xn ). If each component of V is continuous, then V is a continuous vector field, and more generally V is a C k vector field if each component of V is k times continuously differentiable. A vector field can be visualized as assigning a vector to individual points within an n-dimensional space. [1] Given two C k -vector fields V , W defined on S and a real
102
103
17.2. EXAMPLES
valued C k -function f defined on S , the two operations scalar multiplication and vector addition
z
(f V )( p) := f ( p)V ( p) (V + W )( p) := V ( p) + W ( p)
define the module of C k -vector fields over the ring of Ck functions.
y
17.1.2 Coordinate transformation law
In physics, a vector is additionally distinguished by how its coordinates change when one measures the same vector with respect to a different background coordinate system. The transformation properties of vectors distinguish a vector as a geometrically distinct entity from a simple list of scalars, or from a covector. Thus, suppose that ( x 1 ,..., xn) is a choice of Cartesian coordinates, in terms of which thecomponents of thevector V are
x A vector field on a sphere
can make sense of the notion of smooth (analytic) vector fields. The collection of all smooth vector fields on a smooth manifold M is often denoted by Γ(TM ) or C ∞ (M ,TM ) (especially when thinking of vector fields as sections); the collection of all smooth vector fields is also denoted by X (M ) (a fraktur “X”).
V x = (V 1,x , . . . , Vn,x )
and suppose that (y1 ,...,yn) are n functions of the xi defin- 17.2 ing a different coordinate system. Then the components of the vector V in the new coordinates are required to satisfy the transformation law
Examples
Such a transformation law is called contravariant. A similar transformation law characterizes vector fields in physics: specifically, a vector field is a specification of n functions in each coordinate system subject to the transformation law (1) relating the different coordinate systems. Vector fields are thus contrasted with scalar fields, which The flow field around an airplane is a vector field in R3 , here viassociate a number or scalar to every point in space, and sualized by bubbles that follow the streamlines showing a wingtip arealso contrasted with simple lists of scalar fields, which vortex . do not transform under coordinate changes. 17.1.3 Vector fields on manifolds
Given a differentiable manifold M , a vector field on M is an assignment of a tangent vector to each point in M .[2] More precisely, a vector field F is a mapping from M into the tangent bundle TM so that p ◦ F is the identity mapping where p denotes the projection from TM to M . In other words, a vector field is a section of the tangent bundle. If the manifold M is smooth or analytic—that is, the change of coordinates is smooth (analytic)—then one
• A vector field for the movement of air on Earth will
associate for every point on the surface of the Earth a vector with the wind speed and direction for that point. This can be drawn using arrows to represent the wind; the length (magnitude) of the arrow will be an indication of the wind speed. A “high” on the usual barometric pressure map would then act as a source (arrows pointing away), anda “low” would be a sink (arrows pointing towards), since air tends to move from high pressureareas to low pressureareas.
• Velocity field of a moving fluid.
In this case, a velocity vector is associated to each point in the
104
CHAPTER 17. VECTOR FIELD
fluid.
• Streamlines, Streaklines and Pathlines are3typesof lines that can be made from vector fields. They are : streaklines — as revealed in wind tunnels using smoke. streamlines (or fieldlines)— as a line depicting the instantaneous field at a given time. pathlines — showing the path that a given particle (of zero mass) would follow.
V =
∇f =
�
�
∂f ∂f ∂f ∂f , , ,..., . ∂x 1 ∂x 2 ∂x 3 ∂x n
The associated flow is called the gradient flow, and is used in the method of gradient descent. The path integral along any closed curve γ (γ(0) = γ(1)) in a conservative field is zero:
⟨
V (x), dx =
γ
⟩
⟨∇
f (x), dx = f (γ (1))
γ
⟩
− f (γ (0)).
where the angular brackets and comma: ⟨, ⟩ denotes the • Magnetic fields. The fieldlines can be revealed using inner product of two vectors (strictly speaking – the integrand V ( x ) is a 1-form rather than a vector in the elesmall iron filings. mentary sense).[4] • Maxwell’s equations allow us to use a given set of initial conditions to deduce, for every point in Euclidean space, a magnitude and direction 17.2.2 Central field for the force experienced by a charged test particle at that point; the resulting vector field is the A C ∞ -vector field over Rn \ {0} is called a central field if electromagnetic field.
• A gravitational fieldgenerated by any massive object
is also a vector field. For example, the gravitational V (T ( p)) = T (V ( p)) (T ∈ O(n, R)) field vectors for a spherically symmetric body would all point towards the sphere’s center with the magni- where O(n, R) is the orthogonal group. We say centude of the vectors reducing as radial distance from tral fields are invariant under orthogonal transformations the body increases. around 0. The point 0 is called the center of the field. 17.2.1 Gradient field Since orthogonal transformations are actually rotations and reflections, the invariance conditions mean that vectors of a central field are always directed towards, or away from, 0; this is an alternate (and simpler) definition. A central field is always a gradient field, since defining it on one semiaxis and integrating gives an antigradient.
17.3 Operations on vector fields 17.3.1
Line integral
Main article: Line integral
A vector field that has circulation about a point cannot be written as the gradient of a function.
Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).[3] A vector field V defined on a set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that
A common technique in physics is to integrate a vector field along a curve, i.e. to determine its line integral. Given a particle in a gravitational vector field, where each vector representstheforce acting on theparticle at a given point in space, the line integral is the work done on the particle when it travels along a certain path. The line integral is constructed analogously to the Riemann integral and it exists if the curve is rectifiable (has finite length) and the vector field is continuous. Given a vector field V and a curve γ parametrized by [a, b] (where a and b are real) the line integral is defined as
105
17.4. HISTORY
� ⟨
b
V (x), dx =
γ
17.3.2
� ⟨
⟩
V (γ (t)), γ ′ (t) dt .
⟩
a
Divergence
Main article: Divergence The divergence of a vector field on Euclidean space is a function (or scalar field). In three-dimensions, the divergence is defined by div F = ∇ · F =
∂F 1 ∂F 2 ∂F 3 + + , ∂x ∂y ∂z
with the obvious generalization to arbitrary dimensions. The divergence at a point represents the degree to which a small volume around the point is a source or a sink for the vector flow, a result which is made precise by the divergence theorem. The divergence can also be defined on a Riemannian manifold, that is, a manifold with a Riemannian metric that measures the length of vectors. 17.3.3
Curl
Main article: Curl (mathematics)
from this sphere to a unit sphere of dimensions n − 1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S n-1. This defines a continuous map from S to Sn-1 . The index of the vector field at the point is the degree of this map. It can be shown that this integer does not depend on thechoice of S, andthereforedepends only on the vector field itself. The index of the vector field as a whole is defined when it has just a finite numberof zeroes. In thiscase, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes. The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). it is equal to +1 around a source, and more generally equal to (−1) k arounda saddle that hask contracting dimensions andn-k expanding dimensions. For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the hairy ball theorem, which states that if a vector in R3 is assigned to each point of the unit sphere S2 in a continuous manner, then it is impossible to “comb the hairs flat”, i.e., to choose the vectors in a continuous way such that they are all non-zero and tangent to S 2 . For a vector field on a compact manifold with a finite number of zeroes, the Poincaré-Hopf theorem states that the index of the vector field is equal to the Euler characteristic of the manifold.
The curl is an operation which takes a vector field and 17.4 produces another vector field. The curl is defined only in three-dimensions, but some properties of the curl can be captured in higher dimensions with the exterior derivative. In three-dimensions, it is defined by
curl F = ∇×F =
�
∂F 3 ∂y
−
∂F 2 ∂z
� −� e1
∂F 3 ∂x
−
∂F 1 ∂z
� �
∂F 2 e2 + ∂x
History
−
∂F 1 ∂y
�
e3 .
The curl measures the density of the angular momentum of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis. This intuitive description is made precise by Stokes’ theorem. 17.3.4
Index of a vector field
The index of a vector field is an integer that helps to describe the behaviour of a vector field around an isolated zero (i.e.,an isolatedsingularityof thefield). In theplane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity. Let the dimension of the manifold on which the vector field is defined be n . Take a small sphere S around the zero so that no other zeros lie in the interior of S. A map
Magnetic field lines of an iron bar (magnetic dipole)
Vector fields arose originally in classical field theory in 19th century physics, specifically in magnetism. They were formalized by Michael Faraday, in his concept of lines of force , who emphasized that the field itself should be an object of study, which it has become throughout physics in the form of field theory. In addition to the magnetic field, other phenomena that were modeled as vector fields byFaradayincludetheelec-
106
CHAPTER 17. VECTOR FIELD
trical field and light field.
17.5
unique solution x(t) = 1/(x0 - t) if x 0 ≠ 0 (and x(t) = 0 for all t if x0 = 0). Hence for x0 ≠ 0, x(t) is undefined at t = x0 so cannot be defined for all values of t.
Flow curves
Main article: Integral curve
17.6
Consider the flow of a fluid through a region of space. At any given time, any point of the fluid has a particular velocity associated with it; thus there is a vector field associated to any flow. The converse is also true: it is possible to associate a flow to a vector field having that vector field as its velocity. Given a vector field V defined on S , one defines curves γ(t ) on S such that for each t in an interval I
The difference between a scalar and vector field is not that “a scalar is just one number while a vector is several numbers”. The difference is in: how their coordinates respond to coordinate transformations. A scalar is a coordinate whereas a vector can be described by coordinates, but it is not the collection of its coordinates. 17.6.1
Difference between scalar and vector field
Example 1
γ ′ (t) = V (γ (t)) .
This example is about 2-dimensional Euclidean space 2 By the Picard–Lindelöf theorem, if V is Lipschitz con- (R ) where we examine Euclidean ( x , y) and polar (r , θ) origin). Thus x tinuous there is a unique C 1-curve γ x for each point x in coordinates (which are undefined at the 2 = r cos θ and y = r sin θ and also r = x 2 + y 2 , cos θ S so that, for some ε > 0, = x /( x 2 + y2 )1/2 and sin θ = y /( x 2 + y 2 )1/2 . Suppose we have a scalar field which is given by the constant function 1, and a vector field which attaches a vector in the γ x (0) = x r -direction with length 1 to each point. More precisely, they are given by the functions γ ′ (t) = V (γ (t)) (t ∈ (−ε, +ε) ⊂ R). x
x
The curves γ x are called integral curves or trajectories (or less commonly, flow lines) of the vector field V and partition S into equivalence classes. It is not always possible to extend the interval (−ε, +ε) to the whole real number line. The flow may for example reach the edge of S in a finite time. In two or three dimensions one can visualize the vector field as giving rise to a flow on S . If we drop a particle into this flow at a point p it will move along the curve γ p in the flow depending on the initial point p . If p is a stationary point of V (i.e., the vector field is equal to the zero vector at the point p ), then the particle will remain at p. Typicalapplications are streamline in fluid, geodesic flow, and one-parameter subgroups and the exponential map in Lie groups. 17.5.1 Complete vector fields
By definition, a vector field is called complete if every one of its flow curves exist for all time. [5] In particular, compactly supported vector fields on a manifold are complete. If X is a complete vector field on M , then the oneparameter group of diffeomorphisms generated by the flow along X exists for all time. On a compact manifold without boundary, every smooth vector field is complete. An example of an incomplete vector field V on the real line R is given by V (x) = x2 . For, the differential equation dx/dt = x2 , with initial condition x(0) = x 0 , has as its
spolar : (r, θ )
→ 1,
vpolar : (r, θ)
→ (1, 0).
Let us convert these fields to Euclidean coordinates. The vector of length 1 in the r -direction has the x coordinate cos θ and the y coordinate sin θ. Thus in Euclidean coordinates the same fields are described by the functions sEuclidean : (x, y ) vEuclidean : (x, y )
→ 1, → (cos θ, sin θ) =
�
x
,
y
√ √ x2 + y 2
x2 + y 2
We see that while the scalar field remains the same, the vector field now looks different. The same holds even in the 1-dimensional case, as illustrated by thenext example. 17.6.2
Example 2
Consider the 1-dimensional Euclidean space R with its standard Euclidean coordinate x . Suppose we have a scalar field and a vector field which are both given in the x coordinate by the constant function 1, sEuclidean : x
→ 1,
vEuclidean : x
→ 1.
�
.
107
17.10. REFERE REFERENCES NCES
17.10 Refer Referenc ences es Thus, we have a scalar field which has the value 1 every- 17.10 where and a vector field which attaches a vector in the x -direction -direction with magnitude 1 unit of x to to each point. [1] Galbis, Antonio & Maestre, Manuel (2012). Vector Analysis Versus Vector Calculus . Springer. p. 12. ISBN 12. ISBN 978Now consider the coordinate ξ := 2 x . If x changes changes one 1-4614-2199-3.. 1-4614-2199-3 unit then ξ changes 2 units. But since we wish the integral of v along a path to be independent of coordinate, [2] Tu, Loring W. W. (2010). “Vector fields”. fields”. An Introduction to this means v*dx=v'*dξ. So from x increase by 1 unit, ξ Manifolds. Springer. p. 149. ISBN 149. ISBN 978-1-4419-7399-3. 978-1-4419-7399-3. increases by 1/2 unit, so v' must be 2. Thus this vector Vectorss and Ve Vector ctor Oper Operators ators. CRC Dawber, r, P.G. P.G. (1987) (1987).. Vector field field has a magnitude magnitude of 2 in units of ξ. Theref Therefore, ore, in [3] Dawbe Press. p. 29. ISBN 29. ISBN 978-0-85274-585-4. 978-0-85274-585-4. the ξ coordinate the scalar field and the vector field are described by the functions [4] T. Frankel (2012), The Geometry of Physics (3rd ed.),
Cambridge University Press, p. xxxviii, ISBN xxxviii, ISBN 978-1107602601
sunusual : ξ
→ 1,
vunusual : ξ
which are different.
17.7 17.7
→ 2
f-re f-rela lated tedne ness ss
Given a smooth a smooth function between function between manifolds, f : M → → N , the derivative the derivative is is an induced map on tangent on tangent bundles, bundles, f **:: TM → → TN . Given vector fields V : M → → TM and and W : N → → TN , we say that W is is f -related to V if if the equation W ∘ ∘ * f = f * ∘ V holds. holds. If V ᵢ is f -related -related to W ᵢ,ᵢ, i = = 1, 2, then the Lie the Lie bracket [ bracket [V 1 , V 2 ] is f -related -related to [W 1 , W 2 ].
17.8 Genera Generaliz lizati ation onss
Differential geometry. Spri [5] Sharpe, R. (1997). Differential Spring nger er-Verlag. ISBN Verlag. ISBN 0-387-94732-9. 0-387-94732-9.
17.11 17.11
Bibli Bibliogr ograph aphy y
• Hubbard, J. H.; H.; Hubbard, B. B. (1999). Vector cal-
culus, linear algebra, and differential forms. A uni fied approach. Upper Upper Saddle River River,, NJ: Prentic Prenticee
Hall. ISBN Hall. ISBN 0-13-657446-7. 0-13-657446-7 .
• Warner, Frank (1983) [1971]. Foundations of dif-
ferentiable ferentiable manifolds manifolds and Lie groups . New New Yor Yorkk-
Berlin: Springer-Verlag. Springer-Verlag. ISBN ISBN 0-387-90894-3. 0-387-90894-3.
• Boothby, William William (1986). An introduction to differ-
entiable manifolds manifolds and Riemannian geometry . Pure
and Applied Applied Mathemat Mathematic ics, s, volume volume 120 (second (second ed.). Orlando, FL: Academic Press. ISBN Press. ISBN 0-12-116053X.
Replacing vectors by p-vectors ( p pth exterior power of vectors) yields p -vector fields; fields; taking the dual the dual space and space and exterior powers yields differential yields differential k -forms, -forms, and combining these yields general tensor general tensor fields. fields. 17.12 17.12 Ex Exte tern rnal al link linkss Algebrai Algebraical cally, ly, vector vector fields fields can be charac characteriz terized ed as derivations of derivations of the algebra of smooth functions on the • Hazewink Hazewinkel, el, Michie Michiel,l, ed. (2001 (2001),), “Vector field”, field”, manifold, which leads to defining a vector field on a comEncyclopedia of Mathematics , Springer Springer,, ISBN 978mutative algebra algebra as a derivation on the algebra, algebra, which is 1-55608-010-4 developed in the theory of differential of differential calculus over commutative algebras. algebras. • Vector field — field — Mathworld Mathworld
17.9 17.9 See See also also
• Eisenbud–Le Eisenbud–Levine–Khims vine–Khimshiash hiashvili vili signatur signaturee form formula ula • Field line • Field strength • Lie derivative • Scalar field • Time-dependent vector field • Vector fields in cylindrical and spherical spherical coordinates • Tensor fields
• Vector field — field — PlanetMath PlanetMath • 3D Magnetic field viewer • Vector fields and field lines • Vector field simulation An simulation An interactive application to show the effects of vector fields
Chapter 18
Vector space This article is about linear (vector) spaces. For the struc- ally endowed with additional structure, which may be a ture in incidence geometry, see Linear see Linear space (geometry). (geometry). topology topology,, allowing the consideration of issues of proximA vector space (also called a linear space ) is a col- ityand continuity continuity.. Among Among these these topol topologi ogies es,, those those that that are defined by a norm a norm or or inner inner product are product are more commonly used, as having a notion of distance of distance between between two vectors. v+w This is particularly the case of of Banach Banach spaces and spaces and Hilbert Hilbert v spaces,, which are fundamental in mathematical analysis. spaces Historically, Historically, the first ideas leading to vector spaces can be w traced back as far as the 17th century’s analytic century’s analytic geometry,, matrices try matrices,, systems of linear of linear equations, equations, and Euclidean v vectors. The modern, more abstract treatment, first forv+2w mulated by Giuseppe by Giuseppe Peano in Peano in 1888, encompasses encompasses more general objects than Euclidean space, but much of the 2w theory can be seen as an extension of classical geometlike lines,, planes planes and and their higher-dimensional higher-dimensional Vector Vector addition and scalar multiplication multiplication:: a vector vector v (blue) is ric ideas like lines added to another vector w (red, upper illustration). Below, w is is analogs. w (red, stretched by a factor of 2, yielding the sum v + 2 w . Today, vector spaces are applied throughout mathematics, science and engineering and engineering.. They They are are the appro appropri pri-lection of objects called vectors, which may be added be added ate linear-algebraic notion to deal with systems of lintogether and and multiplied multiplied (“scaled”) by numbers, called ear equation equationss; offer a framework for Fourier for Fourier expansion, expansion, scalars in in this context. context. Scalars Scalars are often often taken taken to be which whichis is empl employ oyed ed in image compress compression ion routines; or proreal numbers, numbers, but there are also vector spaces with scalar vide an environment that can be used for solution techmultiplication by complex by complex numbers, numbers , rational numbers, numbers, or niques for partial for partial differential equations. equations. Furthe Furthermo rmore, re, generally any field any field.. The operations of vector addition and vector spaces furnish an abstract, coordinate-free way scalar multiplication must satisfy certain requirements, of dealing with geometrical and physical objects such called axioms, listed below listed below.. as tensors as tensors.. This in turn allows the examination of local of manifolds by by linearization techniques. VecEuclidean vectors are vectors are an example of a vector space. They properties of manifolds represent physical quantities such as forces: forces: any any two two tor spaces may be generalized in several ways, leading to forces(ofthesametype)canbeaddedtoyieldathird,and more advanced notions in geometry and abstract and abstract algebra. algebra. the multiplication of a force a force vector by vector by a real multiplier is another another force force vector. vector. In the same vein, vein, but in a more geometric sense, geometric sense, vectors representing displacements in 18.1 Introducti Introduction on and definiti definition on the plane or in three-dimensional in three-dimensional space also space also form vector spaces. Vectors Vectors in vector spaces do not necessarily necessarily have to be arrow-like objects as they appear in the mentioned The concept of vector space will first be explained by deexamples: examples: vectors are regarded as abstract mathematical mathematical scribing two particular examples: objects with particular properties, which in some cases can be visualize v isualizedd as arrows. 18.1.1 First First examp example: le: arrows arrows in the plane plane Vector spaces are the subject of linear of linear algebra and algebra and are well characterized by their dimension dimension,, which, roughly The first example of a vector space consists of arrows arrows in in speaking, specifies the number of independent directions a fixed plane fixed plane,, starting at one fixed point. This is used in in the space. Infinite-dimensional vector spaces arise nat- physics to describe forces describe forces or or velocities velocities.. Given Given any any two two urally urally in mathemati mathematical cal analys analysis is,, as func function tion spaces spaces,, whose such arrows, v and w, the parallelogram the parallelogram spanned spanned by these vectors are functions are functions.. These These vector vector spaces spaces are are genergener- two arrows contains one diagonal arrow that starts at the 108
109
18.1. INTRODUCTIO INTRODUCTION N AND DEFINIT DEFINITION ION
origin, too. This new arrow is called the sum of the two arrows and is denoted v + w. In the special case of two arro arrows ws on the the same same line line,, thei theirr sum sum is the the arro arrow w on this this line line whose length is the sum or the difference difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive positive real real number a, the arrow that has the same direction as v , but is dilated or shrunk by multiplying its length by a, is called multiplication of v by a. It is denoted av. When a is negative, av is defined as the arrow pointing in the opposite direction, instead. The following shows a few examples: if a = 2, the resulting vector vector aw has the same same direc directio tionn as w, but is stretc stretche hedd to the double length of w (right image below). Equivalently 2w is the sum w + w . Moreover, (−1)v = −v has the opposite direction and the same length as v (blue vector pointing down in the right image).
To qualify as a vector space, the set V and and the operations of addition and multiplication must adhere to a number of requirements called axioms called axioms..[1] In the list below, let u, v and w be arbitrary vectors in V , and a and b scalars in F . These axioms generalize properties properties of the vectors introduced in the above examples. examples. Indeed, the result of additionoftwoorderedpairs(asinthesecondexampleabove) does not depend on the order of the summands: ( x x ᵥ, yᵥ) + ( x x , y) = ( x x , y) + ( x x ᵥ, yᵥ).
Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both exampl examples. es. Thus, by disrega disregardin rdingg the concrete concrete nature nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of 18.1.2 18.1.2 Secon Second d examp example: le: order ordered ed pairs pairs of of vector space. numbers Subtraction of two vectors and division by a (non-zero) A second key example of a vector space is provided by scalar can be defined as pairs of real numbers x and and y. (The order of the compov − w = v + (−w), nents x and and y is significant, so such a pair is also called an ordered pair.) pair.) Such a pair is written written as ( x x , y). The sum of v/a = (1/a)v. two such pairs and multiplication of a pair with a number is defined as follows: When the scalar field F is the real the real numbers R , the vector space is called a real vector space . When the scalar ( x x 1 , y1 ) + ( x x 2 , y2 ) = ( x x 1 + x 2 , y1 + y2 ) field is the complex the complex numbers, numbers , it is called a complex vector space . These two cases are the ones used most often in engineering. The general definition of a vector space and allows scalars to be elements of any fixed field F . The notion is then known as an F -vector spaces or a vector a ( x x , y) = (ax , ay). space over F . A field is, essentially, a set of numbers possessing addition sessing addition,, subtraction subtraction,, multiplication multiplication and and division division The first example above reduces to this one if the arrows operations. [nb 3] For example, rational example, rational numbers also numbers also form are represented by the pair of Cartesian of Cartesian coordinates of coordinates of a field. their end points. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, there is, in general 18.1.3 18.1.3 Defini Definiti tion on vector spaces, no notion of nearness of nearness,, angles angles or or distances distances.. To deal with such matters, particular types of vector A vector space over a field a field F is is a set a set V together together with two spaces are introduced; see below see below.. operations operations that satisfy satisfy the eight eight axioms listed below. Elements of V are are commonly called vectors. Elements Elements of F 18.1.4 Altern Alternati ative ve formula ormulati tion onss and eleeleare commonly called scalars. The first operation, called 18.1.4 vector addition or simply addition, takes any two vectors mentary consequences v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two Vector Vector addition and scalar multiplication multiplication are operations, vectors. The second operation, called scalar multiplica- satisfying the closure the closure property: property: u + v and av are in V for for tion takes any scalar a and any vector v and gives another all a in F ,and , and u, v in V . Some Some olde olderr sourc sources es menti mention on these these vector av. properties properties as separate axioms.[2] In this article, vectors are distinguished from scalars by In the parlance of abstract of abstract algebra, algebra, the first four axioms [nb 1] boldface. In the two examples above, the field is the can be subsumed by requiring the set of vectors to be fieldoftherealnumbersandthesetofthevectorsconsists an abelian an abelian group under group under addition. The remaining axioms of the planar arrows with fixed starting point and of pairs give this group an F -module module struc structure. ture. In other words, words, of real numbers, respectively. there is a ring a ring homomorphism f from from the field F into into the
110 endomorphism ring of the group of vectors. Then scalar multiplication av is defined as ( f (a))(v).[3] There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to theadditive group of vectors: for example the zero vector 0 of V and the additive inverse −v of any vector v are unique. Other properties follow from the distributive law, for example av equals 0 if and only if a equals 0 or v equals 0.
CHAPTER 18. VECTOR SPACE
18.3
Examples
Main article: Examples of vector spaces
18.3.1 Coordinate spaces
Main article: Coordinate space
The most simple example of a vector space over a field F is the field itself, equipped with its standard addition 18.2 History and multiplication. More generally, a vector space can be composed of n-tuples (sequences of length n) of elements Vector spaces stem from affine geometry via the intro- of F , such as duction of coordinates in the plane or three-dimensional (a1 , a2 , ..., an), where each ai is an element of space. Around 1636, Descartes and Fermat founded F .[13] analytic geometry by equating solutions to an equation of two variables with points on a plane curve.[4] In 1804, to achieve geometric solutions without using coordinates, A vector space composed of all the n-tuples of a field F Bolzano introduced certain operations on points, lines is known as a coordinate space , usually denoted F n . The andplanes, whicharepredecessorsof vectors.[5] Hiswork case n = 1 is the above-mentioned simplest example, in was then used in the conception of barycentric coordi- which the field F is also regarded as a vector space over nates by Möbius in 1827.[6] In 1828 C. V. Mourey sug- itself. The case F = R and n = 2 was discussed in the gested the existence of an algebra surpassing not only or- introduction above. dinary algebra but also two-dimensional algebra created by him searching a geometrical interpretation of complex 18.3.2 Complex numbers and other field numbers.[7] extensions The definition of vectors was founded on Bellavitis' notion of the bipoint, an oriented segment of which one end The set of complex numbers C, i.e., numbers that can be is the origin andtheother a target, then further elaborated written in the form x + iy for real numbers x and y where with the presentation of complex numbers by Argand i is the imaginary unit, form a vector space over the reals and Hamilton and the introduction of quaternions and with the usual addition and multiplication: ( x + iy) + (a + biquaternions by the latter. [8] They are elements in R 2 , ib) = ( x + a) + i (y + b) and c ⋅ ( x + iy) = ( c ⋅ x ) + i (c ⋅ y) R4 , and R 8 ; their treatment as linear combinations can for real numbers x , y, a, b and c . The various axioms of a be traced back to Laguerre in 1867, who also defined vector space follow from the fact that the same rules hold systems of linear equations. for complex number arithmetic. In 1857, Cayley introduced matrix notation, which allows In fact, theexample of complex numbers is essentially the for a harmonization and simplification of linear maps. same (i.e., it is isomorphic ) to the vector space of ordered Around the same time, Grassmann studied the barycen- pairs of real numbers mentioned above: if we think of the tric calculus initiated by Möbius. He envisaged sets of complex number x + i y as representing the ordered pair abstract objects endowed with operations. [9] In his work, ( x , y) in the complex plane then we see that the rules for the concepts of linear independence and dimension, as sum and scalar product correspond exactly to those in the well as scalar products, are present. In fact, Grassmann’s earlier example. 1844 work exceeds the framework of vector spaces, since his consideration of multiplication led him to what are to- More generally, field extensions provide another class of daycalled algebras. Peano wasthefirsttogivethemodern examples of vector spaces, particularly in algebra and definition of vector spaces and linear maps in 1888. [10] algebraic number theory: a field F containing a smaller field E is an E -vector space, by the given multiplication An important development of vector spaces is due to the and addition operations of F .[14] For example, the comconstruction of function spaces by Lebesgue. This was plex numbers are a vector space over R, and the field exlater formalized by Banach and Hilbert, around 1920.[11] tension Q(i√ 5) is a vector space over Q. At that time, algebra and the new field of functional analysis began to interact, notably with key concepts such as spaces of p-integrable functions and Hilbert spaces.[12] 18.3.3 Function spaces Vector spaces, including infinite-dimensional ones, then became a firmly established notion, and many mathemat- Functions from any fixed set Ω to a field F also form vecical branches started making use of this concept. tor spaces, by performing addition and scalar multiplica-
18.4. BASIS AND DIMENSION
111
tion pointwise. That is, the sum of two functions f and g is the function ( f + g) given by ( f + g)(w) = f (w) + g(w), and similarly for multiplication. Such function spaces occur in many geometric situations, when Ω is the real line or an interval, or other subsets of R . Many notions in topology and analysis, such as continuity, integrability or differentiability are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. [15] Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of functional analysis, see below. Algebraic constraints also yield vector spaces: the vector space F [x] is given by polynomial functions: f ( x ) = r 0 + r 1 x + ... + rn ₋₁ x n−1 + rnx n , where the coefficients r 0 , ..., rn are in F .[16]
18.3.4
A vector v in R2 (blue) expressed in termsof different bases: using the standard basis of R2 v = x e1 + y e2 (black), and using a different, non-orthogonal basis: v = f 1 + f 2 (red).
Linear equations
Main articles: Linear equation, Linear differential called coordinates or components. A basis is a (finite or infinite) set B = { bi }i ∈ I of vectors b i , for convenience equation, and Systems of linear equations often indexed by some index set I , that spans the whole space and is linearly independent. “Spanning the whole Systems of homogeneous linear equations are closely tied space” meansthat any vector v canbeexpressedasafinite to vector spaces.[17] For example, the solutions of sum (called a linear combination ) of the basis elements:
are given by triples with arbitrary a , b = a/2, and c = −5a/2. They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. Matrices can be used to condense multiple linear equations as above into one vector equation, namely
where the ak are scalars, called the coordinates (or the components) of the vector v with respect to the basis B, and bik (k = 1, ..., n) elements of B. Linear independence means that the coordinates ak are uniquely determined for any vector in the vector space. For example, the coordinate vectors e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), to en = (0, 0, ..., 0, 1), form a basis of Ax = 0, F n , called the standard basis, since any vector ( x 1 , x 2 , ..., xn) can be uniquely expressed as a linear combination of 1 3 1 where A = 4 2 2 is the matrix containing the coef- these vectors: ficients of the given equations, x is the vector ( a, b, c ), Ax denotes the matrix product, and 0 = (0, 0) is the zero vec( x 1 , x 2 , ..., xn) = x 1 (1, 0, ..., 0) + x 2(0, 1, 0, ..., tor. In a similar vein, thesolutionsof homogeneous linear 0) + ... + xn (0, ..., 0, 1) = x 1 e1 + x 2 e2 + ... + differential equations form vector spaces. For example, xnen.
�
�
f ′′( x ) + 2 f ′( x ) + f ( x ) = 0
The corresponding coordinates x 1 , x 2 , ..., xn are just the Cartesian coordinates of the vector. yields f ( x ) = a e− x + bx e− x , where a and b are arbitrary Every vector space has a basis. This follows from constants, and e x is the natural exponential function. Zorn’s lemma, an equivalent formulation of the Axiom of Choice.[18] Given the other axioms of Zermelo–Fraenkel set theory, the existence of bases is equivalent to the ax18.4 Basis and dimension iom of choice.[19] The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a Main articles: Basis and Dimension given vector space have the same number of elements, or Bases allow to represent vectors by a sequence of scalars cardinality (cf. Dimension theorem for vector spaces ).[20]
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CHAPTER 18. VECTOR SPACE
It is called the dimension of the vector space, denoted dim V . If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory.[21] The dimension of the coordinate space F n is n , by the basis exhibited above. The dimension of the polynomial ring F [ x ] introduced above is countably infinite, a basis is given by 1, x , x 2 , ... A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.[nb 4] Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation.[22] For example, the solution space for the above equation is generated by e − x and xe − x . These two functions are linearly independent over R, so the dimension of this space is two, as is the degree of the equation. A field extension over the rationals Q can be thought of as a vector space over Q (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of Q , and otherwise ignoring the field multiplication). The dimension (or degree) of the field extension Q(α) over Q depends on α. If α satisfies some polynomial equation qnαn
+ qn ₋₁αn−1
+ ... + q 0 = 0, with rational coefficients qn, ..., q0 . ("αis algebraic"), the dimension is finite. More precisely, it equals the degree of the minimal polynomial having α as a root.[23] For example, the complex numbers C are a two-dimensionalreal vector space, generated by1 andthe imaginaryunit i . The latter satisfies i 2 +1=0,anequation of degree two. Thus, C is a two-dimensional R -vector space (and, as any field,one-dimensionalas a vector space overitself, C).If α isnot algebraic, the dimension of Q(α) over Q is infinite. For instance, for α = π there is no such equation, in other words π is transcendental.[24]
18.5 Linear maps and matrices Main article: Linear map The relation of two vector spaces can be expressed by linear map or linear transformation . They are functions that reflect the vector space structure—i.e., they preserve sums and scalar multiplication: f (x + y ) = f (x) + f (y) and f (a · x ) = a · f (x) for all x and y in V , all a in F .[25]
An isomorphism is a linear map f : V → W such that there exists an inverse map g : W → V , which is a map such that the two possible compositions f ∘ g : W → W
and g ∘ f : V → V are identity maps. Equivalently, f is both one-to-one (injective) and onto (surjective).[26] If there exists an isomorphism between V and W , the two spaces are said to be isomorphic ; they are then essentially identical as vector spaces, since all identities holding in V are, via f , transported to similar onesin W , and vice versa via g.
Describing an arrow vector v by its coordinates x and y yields an isomorphism of vector spaces.
For example, the “arrows in the plane” and “ordered pairs of numbers” vector spaces in theintroduction areisomorphic: a planar arrow v departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the x - and y-component of the arrow, as shown in the image at the right. Conversely, given a pair ( x , y), the arrow going by x to the right (or to the left, if x is negative), and y up (down, if y is negative) turns back the arrow v. Linear maps V → W between two vector spaces form a vector space HomF (V , W ),alsodenotedL( V , W ).[27] The space of linear maps from V to F is called the dual vector space, denoted V ∗ .[28] Via the injective natural map V → V ∗∗ , any vector space can be embedded into its bidual ; the map is an isomorphism if and only if the space is finitedimensional. [29] Once a basis of V is chosen, linear maps f : V → W are completely determined by specifying the images of the basis vectors, because any element of V is expressed uniquely as a linear combination of them. [30] If dim V = dim W , a 1-to-1 correspondence between fixed bases of V and W gives rise to a linear map that maps any basis element of V to the corresponding basis element of W . It is an isomorphism, by its very definition. [31] Therefore, two vector spaces areisomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is completely classified (up to isomorphism) by its dimension, a single number. In particular, any n-dimensional F -vector space V is isomorphic to F n . There is, however, no “canonical” or preferred isomorphism; actually an isomorphism φ : F n → V is equivalent to the choice of a basis of V , by mapping the stan-
113
18.6. BASIC CONSTRUCTIONS
dard basis of F n to V , via φ . The freedom of choosing a convenient basis is particularly useful in the infinitedimensional context, see below. 18.5.1
Matrices
Main articles: Matrix and Determinant Matrices are a useful notion to encode linear maps. [32]
ai,j
j changes
n columns
m rows i c h a n g e s
a1,1
a1,2
a1,3 . . .
a2,1
a2,2
a2,3 . . .
a3,1
a3,2
a3,3 . . .
. . .
. . .
. . .
The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r 1 , r 2 , and r3 .
. . .
A typical matrix
They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix A gives rise to a linear map from F n to F m , by the following =
x
�∑ ∑ ∑ n j =1
a1j xj ,
, where
(x1 , x2 ,
··· , x ) =1 a2 x , ··· ,
n j
j
n
j
denotes summation,
∑
n j =1
→
amj xj
�
or, using the matrix multiplication of the matrix A with the coordinate vector x: x ↦ Ax.
with their image under f , f (v). Any nonzero vector v satisfying λv = f (v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ .[nb 5][35] Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V ). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to det( f − λ · Id) = 0. By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in λ, called the characteristic polynomial of f .[36] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.[nb 6] The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f ) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.
Moreover, after choosing bases of V and W , any linear map f : V → W is uniquely represented by a matrix via this assignment.[33] The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.[34] The linear transformation of Rn corresponding to a real n -by-n matrix is orientation pre18.6 serving if and only if its determinant is positive.
Basic constructions
In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to Main article: Eigenvalues and eigenvectors the definitions given below, they are also characterized by universal properties, which determine an object X by Endomorphisms, linear maps f : V → V , are particularly specifying the linear maps from X to any other vector important since in this case vectors v can be compared space. 18.5.2 Eigenvalues and eigenvectors
114 18.6.1 Subspaces and quotient spaces
Main articles: Linear subspace and Quotient vector space A nonempty subset W of a vector space V that is closed
CHAPTER 18. VECTOR SPACE
statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms) V / ker( f ) ≡ im( f ).
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups. An important example is the kernel of a linear map x ↦ Ax for some fixed matrix A, as above. The kernel of this map is the subspace of vectors x such that Ax = 0, which is precisely the set of solutions to the system of homogeneous linear equations belonging to A. This concept also extends to linear differential equations 2
df a0 f +a1 dx +a2 ddxf 2 +
···+a
dn f n dxn
= 0 , where
the coefficients ai are functions in x , too. A line passing through the origin (blue, thick) in R 3 is a linear subspace. It is the intersection of two planes (green and yellow).
under addition and scalar multiplication (and therefore contains the 0-vector of V ) is called a linear subspace of V , or simply a subspace of V , when the ambient space is unambiguously a vector space.[37][nb 7] Subspaces of V are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set S of vectors is called its span, and it is the smallest subspace of V containing theset S . Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of S .[38] A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane . In a vector space of finite dimension n, a vector hyperplane is thus a subspace of dimension n – 1. The counterpart to subspaces are quotient vector spaces.[39] Given any subspace W ⊂ V , the quotient space V /W (" V modulo W ") is defined as follows: as a set, it consists of v + W = {v + w : w ∈ W }, where v is an arbitrary vector in V . The sum of two such elements v1 + W and v2 + W is (v1 + v2 ) + W , and scalar multiplication is given by a · (v + W ) = (a · v ) + W . The key point in this definition is that v1 + W = v2 + W if and only if the difference of v1 and v2 lies in W .[nb 8] This way, the quotient space “forgets” information that is contained in the subspace W . The kernel ker( f ) of a linear map f : V → W consists of vectors v that are mapped to 0 in W .[40] Both kernel and image im( f ) = { f (v) : v ∈ V } are subspaces of V and W , respectively.[41] The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field F ) is an abelian category, i.e. a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups.[42] Because of this, many
In the corresponding map n
→ D(f ) =
f
� i=0
ai
di f dxi
the derivatives of the function f appear linearly (as opposed to f ′′( x )2 , for example). Since differentiation is a linear procedure (i.e., ( f + g)′ = f ′ + g ′ and (c · f )′ = c · f ′ for a constant c ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation D ( f ) = 0 form a vector space (over R or C). 18.6.2 Direct product and direct sum
Main articles: Direct product and Direct sum of modules The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The direct product i∈I V i of a family of vector spaces Vi consists of the set of all tuples ( vi )i ∈ I , which specify for each index i insome index set I an element vi of Vi .[43] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum ⊕i∈I V i (also called coproduct and denoted i∈I V i ), where only tuples with finitely many nonzero vectors are allowed. If the index set I is finite, the two constructions agree, but in general they are different.
∏
⨿
18.6.3
Tensor product
Main article: Tensor product of vector spaces The tensor product V ⊗ F W , or simply V ⊗ W , of two vector spaces V and W is one of the central notions of
115
18.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE
multilinear algebra which deals with extending notions such as linear maps to several variables. A map g : V × W → X is called bilinear if g is linear in both variables v and w. That is to say, for fixed w the map v ↦ g(v, w) is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a universal recipient of bilinear maps g , as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors v1 ⊗ w1 + v2 ⊗ w2 + ... + vn ⊗ wn,
subject to the rules a · (v ⊗ w) = (a · v) ⊗ w = v ⊗ (a · w), where a
is a scalar, (v1 + v2 ) ⊗ w = v1 ⊗ w + v2 ⊗ w, and v ⊗ (w1 + w2 ) = v ⊗ w1 + v ⊗ w2 .[44]
terms to be added. Therefore, the needs of functional analysis require considering additional structures. A vector space may be given a partial order ≤, under which some vectors can be compared. [46] For example, n-dimensional real space R n can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions f = f + − f − ,
where f + denotes thepositive part of f and f − the negative part.[47] 18.7.1
Normed vector spaces and inner product spaces
Main articles: Normed vector space and Inner product space “Measuring” vectors is done by specifying a norm, a datum which measures lengths of vectors, or by an inner product, which measures angles between vectors. Norms and inner products are denoted |v| and ⟨v, w⟩ , respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm |v| := ⟨v, v⟩ . Vector spaces endowed with such data are known as normed vector spaces and inner product [48] Commutativediagram depicting the universal property of the ten- spaces, respectively. sor product. Coordinate space F n can be equipped with the standard dot product: These rules ensure that the map f from the V × W to V ⊗ W that maps a tuple ( v, w) to v ⊗ w is bilinear. The universality states that given any vector space X and ⟨x, y⟩ = x · y = x y + ··· + x y . 1 1 n n any bilinear map g : V × W → X , there exists a unique map u, shown in the diagram with a dotted arrow, whose In R 2 , this reflects the common notion of the angle becomposition with f equals g: u (v ⊗ w) = g(v, w).[45] This tween two vectors x and y, by the law of cosines: is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps x · y = cos (∠(x, y)) · |x| · |y|. from or to this object. Because of this, two vectors satisfying ⟨x, y⟩ = 0 are called orthogonal. An important variant of the standard 18.7 Vector spaces with additional dot product is used in Minkowski space: R4 endowed with the Lorentz product structure
√
From thepoint of view of linearalgebra, vector spaces are completelyunderstoodinsofar as anyvector space is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since theaddition operation allows only finitelymany
⟨x|y⟩ = x1y1 + x2y2 + x3y3 − x4y4. [49] In contrast to the standard dot product, it is not positive definite: ⟨x|x⟩ also takes negative values, for example for x = (0, 0, 0, 1) . Singling out the fourth coordinate—corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity.
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CHAPTER 18. VECTOR SPACE
18.7.2 Topological vector spaces
Main article: Topological vector space Convergence questions are treated by considering vector spaces V carrying a compatible topology, a structure that allows one to talk about elements being close to each other.[50][51] Compatible here means that addition andscalar multiplication have to be continuous maps. Roughly, if x and y in V , and a in F vary by a bounded amount, then so do x + y and ax.[nb 9] To make sense of specifying the amount a scalar changes, the field F also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such topological vector spaces one can consider series of vectors. The infinite sum ∞
�
f i
i=0
denotes the limit of the corresponding finite partial sums of the sequence ( fi )i ∈N of elements of V . For example, the fi could be (real or complex) functions belonging to some function space V , in which case the series is a function series. The mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples.
2
vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem.[52] In contrast, the space of all continuous functions on [0,1] with the same topology is complete. [53] A norm gives rise to a topology by defining that a sequence of vectors v n converges to v if and only if limn→∞|vn − v| = 0. Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of functional analysis—focusses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.[54] The image at the right shows the equivalence of the 1-norm and ∞-norm on R2 : as the unit “balls” enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functionals) V → W , maps between topological vector spaces are required to be continuous. [55] In particular, the (topological) dual space V ∗ consists of continuous functionals V → R (or to C ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.[56] Banach spaces
1
Main article: Banach space ∞
Banach spaces , introduced by Stefan Banach, are com-
plete normed vector spaces.[57] A first example is the vector space ℓ p consisting of infinite vectors with real entries x = ( x 1 , x 2 , ...) whose p-norm (1 ≤ p ≤ ∞) given by
|x|
p
2
Unit “spheres” in R consist of plane vectors of norm 1. Depicted are the unit spheres in different p-norms , for p = 1,2, √ and ∞. The bigger diamond depicts points of 1-norm equal to 2 .
:= (
supi |xi |
∑| | i
1/ p
xi p )
for p < ∞ and |x|∞ :=
is finite. The topologies on the infinite-dimensional space ℓ p are inequivalent for different p. E.g. the sequence of vectors x n = (2 −n , 2 −n , ..., 2−n , 0, 0, ...), i.e. the first 2 n A way to ensure the existence of limits of certain in- components are 2−n , the following ones are 0, converges finite series is to restrict attention to spaces where any to the zero vector for p = ∞, but does not for p = 1: Cauchy sequence has a limit; such a vector space is called |xn|∞ = 2sup(2−n, 0 ) = 2−n → 0 , but complete. Roughly, a vector space is complete provided |xn|1 = i=1 2−n = 2 n · 2−n = 1 . that it contains all necessary limits. For example, the
∑
n
18.7. VECTOR SPACES WITH ADDITIONAL STRUCTURE
117
More generally than sequences of real numbers, functions analysis, in the guise of the Taylor approximation, estab f : Ω → R are endowed with a norm that replaces the lished an approximation of differentiable functions f by above sum by the Lebesgue integral polynomials. [62] By the Stone–Weierstrass theorem, every continuous function on [a, b] can be approximated as closely as desired by a polynomial. [63] A similar ap1/ p proximation technique by trigonometric functions iscom|f | p := |f (x)| p dx . monly called Fourier expansion, and is much applied in Ω engineering, see below. More generally, and more conThe space of integrable functions on a given domain ceptually, the theorem yields a simple description of what Ω (for example an interval) satisfying | f |p < ∞, and “basic functions”, or, in abstract Hilbert spaces, what baequipped with this norm are called Lebesgue spaces, de- sic vectors suffice to generate a Hilbert space H , in the noted L p (Ω).[nb 10] These spaces are complete. [58] (If one sense that the closure of their span (i.e., finite linear comuses the Riemann integral instead, the space is not com- binations and limits of those) is the whole space. Such plete, which may be seen as a justification for Lebesgue’s a set of functions is called a basis of H , its cardinalintegration theory.[nb 11] ) Concretely this means that for ity is known as the Hilbert space dimension.[nb 13] Not any sequence of Lebesgue-integrable functions f 1, f 2 , ... only does the theorem exhibit suitable basis functions as with | fn | p < ∞, satisfying the condition sufficient for approximation purposes, but together with the Gram-Schmidt process, it enables one to construct a basis of orthogonal vectors.[64] Such orthogonal bases are the Hilbert space generalization of the coordinate axes in |f k (x) − f n(x)| p dx = 0 lim k, n→∞ Ω finite-dimensional Euclidean space. there exists a function f ( x ) belonging to the vector space The solutions to various differential equations can be L p (Ω) such that interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular p physical properties are used as basis functions, often |f (x) − f k (x)| dx = 0. lim k→∞ Ω orthogonal.[65] As an example from physics, the timeImposing boundedness conditions not only on the func- dependent Schrödinger equation in quantum mechanics tion, but also on its derivatives leads to Sobolev spaces.[59] describes the change of physical properties in time by means of a partial differential equation, whose solutions are called wavefunctions.[66] Definite values for physical Hilbert spaces properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and Main article: Hilbert space the associated wavefunctions are called eigenstates. The Complete inner product spaces are known as Hilbert spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.[67]
��
�
�
�
18.7.3 Algebras over fields The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).
spaces, in honor of David Hilbert.[60] The Hilbert space L2 (Ω), with inner product given by
⟨f , g⟩ =
�
f (x)g(x) dx,
Ω
where g (x) denotes the complex conjugate of g( x ),[61][nb 12] is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions fn with desirable properties that approximates a given limit function, is equally crucial. Early
Main articles: Algebra over a field and Lie algebra General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an algebra over a field .[68] Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone– Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings andtheir quotientsformthebasis of algebraic geometry, because they are rings of functions of algebraic geometric objects.[69]
118
CHAPTER 18. VECTOR SPACE
18.8 Applications Vector spaces have manifold applications as they occur in many circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods.[73] Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.[74] 18.8.1 A hyperbola , given by the equation x y = 1. The coordinate ring of functions on this hyperbola is given by R[x , y] / ( x · y − 1), an infinite-dimensional vector space over R.
Distributions
Main article: Distribution
A distribution (or generalized function ) is a linear map assigning a number to each “test” function, typically a Lie algebras Another crucial example are , which are nei- smooth function with compact support, in a continuous ther commutative nor associative, but the failure to be so way: in the above terminology the space of distributions is limited by the constraints ([ x , y] denotes the product of is the (continuous) dual of the test function space. [75] The x and y): latter space is endowed with a topology that takes into account not only f itself, but also all its higher derivatives. A standard example is theresult of integrating a test func• [ x , y] = −[y, x ] (anticommutativity), and tion f over some domain Ω:
• [ x , [y[70] , z]] + [y, [ z, x ]] + [ z, [ x , y]] = 0 (Jacobi identity).
I (f ) =
�
f (x) dx.
Ω
Examples include the vector space of n-by-n matrices, with [ x , y ] = xy − yx , the commutator of two matrices, and R3 , endowed with the cross product. The tensor algebra T(V ) is a formal way of adding products to any vector space V to obtain an algebra.[71] As a vector space, it is spanned by symbols, called simple tensors v1 ⊗ v2 ⊗ ... ⊗ vn, where the degree n varies.
When Ω = { p}, the set consisting of a single point, this reduces to the Dirac distribution, denoted by δ, which associates to a test function f its value at the p: δ( f ) = f ( p). Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore, the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, Green’s functions and fundamental solutions are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (e.g., using the Lax– Milgram theorem, a consequence of the Riesz representation theorem).[76]
The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced above. In general, there are no relations between v1 ⊗ v2 and v2 ⊗ v1 . Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing v1 ⊗ v2 = − v2 ⊗ v1 18.8.2 Fourier analysis yields the exterior algebra.[72] When a field, F is explicitly stated, a common term used Main article: Fourier analysis is F -algebra. Resolving a periodicfunction intoasumof trigonometric
119
18.9. GENERALIZATIONS
speech encoding, image compression.[86] The JPEG image format is an application of the closely related discrete cosine transform.[87] The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform.[88] It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences.[89] They in turn are applied in digital filters[90] and as a rapid multiplication algorithm for polynomials and large integers (SchönhageStrassen algorithm).[91][92] The heat equation describes the dissipation of physical properties over time, such as the decline of the temperature of a hot body placed in a colder environment (yellow depicts colder re- 18.8.3 gions than red).
Differential geometry
Main article: Tangent space functions forms a Fourier series , a technique much used The tangent plane to a surface at a point is naturally a in physics and engineering. [nb 14][77] The underlying vector space is usually the Hilbert space L2 (0, 2π), for which the functions sin mx and cos mx (m an integer) form an orthogonalbasis.[78] The Fourierexpansion ofan L2 function f is a0
2
∞
+
�
[am cos (mx) + bm sin (mx)] .
m=1
The coefficients am and bm are called Fourier coefficients of f , and are calculated by the formulas [79]
∫ ∫ =
am 1 π
2π 0
1
π
2π 0
f (t) cos(mt) dt
,
bm
=
f (t) sin(mt) dt.
In physical terms the function is represented as a superposition of sine waves and the coefficients give information about the function’s frequency spectrum.[80] A complex-number form of Fourier series is also commonly used.[79] The concrete formulae above are consequences of a more general mathematical duality called Pontryagin duality.[81] Applied to the group R , it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice encoding positions of atoms in crystals.[82] Fourier series are used to solve boundary value problems in partial differential equations.[83] In 1822, Fourier first used this technique to solve the heat equation.[84] A discreteversion of theFourier series can be used in sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier seriesis finite and its value is equal to the sampled values at all points.[85] The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar,
The tangent space to the 2-sphere at some point is the infinite plane touching the sphere in this point.
vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point. [nb 15] Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceivedof as an abstract vector spacerather than a real coordinate space. The tangent space is the generalization to higher-dimensional differentiable manifolds.[93] Riemannian manifolds are manifolds whose tangent spaces are endowed with a suitable inner product.[94] Derived therefrom, the Riemann curvature tensor encodes all curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time.[95][96] The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.[97]
18.9 Generalizations
120
CHAPTER 18. VECTOR SPACE
18.9.1
Vector bundles
and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional Main articles: Vector bundle and Tangent bundle real division algebras: R, C , the quaternions H and the A vector bundle is a family of vector spaces parametrized octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms. 1
S
18.9.2
Modules
Main article: Module U
Modules are to rings what vector spaces are to fields: the same axioms, applied to a ring R instead of a field F ,
yield modules. [101] The theory of modules, compared to U x R that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. U For example, modules need not have bases, as the Zmodule (i.e., abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as A Möbius strip. Locally, it looks like U × R. free modules. Nevertheless, a vector space can be com[93] continuously by a topological space X . More precisely, pactly defined as a module over a ring which is a field a vector bundle over X is a topological space E equipped with the elements being called vectors. Some authors use the term vector space to mean modules over a division with a continuous map ring.[102] The algebro-geometric interpretation of commutative rings via their spectrum allows the development π : E → X of concepts such as locally free modules, the algebraic −1 such that for every x in X , the fiber π ( x ) is a vector counterpart to vector bundles. space. The case dim V = 1 is called a line bundle. For any vector space V , the projection X × V → X makes the product X × V into a “trivial” vector bundle. Vec- 18.9.3 Affine and projective spaces tor bundles over X are required to be locally a product of X and some (fixed) vector space V : for every x in X , Main articles: Affine space and Projective space there is a neighborhood U of x such that the restriction of Roughly, affine spaces are vector spaces whose origins π to π−1 (U ) is isomorphic[nb 16] to the trivial bundle U × V → U . Despite their locally trivial character, vector bundles may (depending on the shapeof the underlying space X ) be “twisted” in the large (i.e., the bundle need not be (globally isomorphic to) the trivial bundle X × V ). For example, the Möbius strip can be seen as a line bundle over the circle S 1 (by identifying open intervals with the real line). It is, however, different from the cylinder S 1 × R, because the latter is orientable whereas the former is not.[98] Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of thecollection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle S 1 is globally isomorphic to S 1 × R, since there is a global nonzero vector An affine plane (light blue) in R 3 . It is a two-dimensional subfield on S 1 .[nb 17] In contrast, by the hairy ball theorem, space shifted by a vector x (red). there is no (tangent) vector field on the 2-sphere S 2 which is everywhere nonzero. [99] K-theory studies the isomor- are not specified. [103] More precisely, an affine space is a phism classes of all vector bundles over some topo- setwitha freetransitive vector space action. In particular, logical space.[100] In addition to deepening topological a vector space is an affine space over itself, by the map 1
121
18.12. FOOTNOTES V × V → V , (v, a) ↦ a + v.
[8] Some authors (such as Roman 2005) choose to start with this equivalence relation and derive the concrete shape of V /W from this.
If W is a vector space, then an affine subspace is a subset of W obtained by translating a linear subspace V by a [9] This requirement implies that the topology gives rise to a fixed vector x ∈ W ; this space is denoted by x + V (it is uniform structure, Bourbaki 1989, ch. II a coset of V in W ) and consists of all vectors of the form x + v for v ∈ V . An important example is the space of [10] The triangle inequality for |−| p is provided by the Minkowski inequality. For technical reasons, in the solutions of a system of inhomogeneous linear equations Ax = b
context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm.
functions in L 2 of Lebesgue measure, being ungeneralizing the homogeneous case b = 0 above.[104] The [11] “Many bounded, cannot be integrated with the classical Riemann space of solutions is the affine subspace x + V where x is integral. So spaces of Riemann integrable functions would a particular solution of the equation, and V is the space not be complete in the L2 norm, and the orthogonal deof solutions of the homogeneous equation (the nullspace composition would not apply to them. This shows one of of A). the advantages of Lebesgue integration.”, Dudley 1989, §5.3, p. 125 The set of one-dimensional subspaces of a fixed finitedimensional vector space V is known as projective space ; [12] For p ≠2, L p (Ω) is not a Hilbert space. it may be used to formalize the idea of parallel lines intersecting at infinity.[105] Grassmannians and flag manifolds [13] A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra above. For distinction, the generalize this by parametrizing linear subspaces of fixed latter is then called a Hamel basis. dimension k and flags of subspaces, respectively.
18.10
See also
• Vector (mathematics and physics), for a list of various kinds of vectors
18.11
Notes
[14] Although the Fourier series is periodic, the technique can be applied to any L2 function on an interval by considering the function to be continued periodically outside the interval. See Kreyszig 1988, p. 601 [15] Thatistosay(BSE-3 2001), the plane passing through the pointofcontactP such that thedistancefrom a point P 1 on the surface to the plane is infinitesimally small compared to the distance from P 1 to P in the limit as P 1 approaches P along the surface.
[16] That is, there is a homeomorphism from π−1 (U ) to V × U [1] It is also common, especially in physics, to denote vectors which restricts to linear isomorphisms between fibers. with an arrow on top:⃗v . [17] A line bundle, such as the tangent bundle of S 1 is triv[2] This axiom refers to two different operations: scalar mulial if and only if there is a section that vanishes nowhere, tiplication: b v; and field multiplication: ab. It does not assee Husemoller 1994, Corollary 8.3. The sections of the sert the associativity of either operation. More formally, tangent bundle are just vector fields. scalar multiplication is the semigroup action of the scalars on the vector space. Combined with the axiom of the identity element of scalar multiplication, it is a monoid 18.12 Footnotes action. [3] Some authors (such as Brown 1991) restrict attention to the fields R or C, but most of the theory is unchanged for an arbitrary field.
[1] Roman 2005, ch. 1, p. 27
[4] The indicator functions of intervals (of which there are infinitely many) are linearly independent, for example.
[3] Bourbaki 1998, §II.1.1. Bourbaki calls the group homomorphisms f (a) homotheties.
[5] The nomenclature derives from German "eigen", which means own or proper.
[4] Bourbaki 1969, ch. “Algèbre linéaire et algèbre multilinéaire”, pp. 78–91
[6] Roman 2005, ch. 8, p. 140. See also Jordan–Chevalley decomposition.
[5] Bolzano 1804
[7] This is typically the case when a vector space is also considered as an affine space. In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it.
[2] van der Waerden 1993, Ch. 19
[6] Möbius 1827 [7] Crowe, Michel J. (1994), A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, Dover, p. 11 and 16, ISBN 0-486-67910-1
122
CHAPTER 18. VECTOR SPACE
[8] Hamilton 1853
[46] Schaefer & Wolff 1999, pp. 204–205
[9] Grassmann 2000
[47] Bourbaki 2004, ch. 2, p. 48
[10] Peano 1888, ch. IX
[48] Roman 2005, ch. 9
[11] Banach 1922
[49] Naber 2003, ch. 1.2
[12] Dorier 1995, Moore 1995
[50] Treves 1967
[13] Lang 1987, ch. I.1
[51] Bourbaki 1987
[14] Lang 2002, ch. V.1
[52] Kreyszig 1989, §4.11-5
[15] e.g. Lang 1993, ch. XII.3., p. 335
[53] Kreyszig 1989, §1.5-5
[16] Lang 1987, ch. IX.1
[54] Choquet 1966, Proposition III.7.2
[17] Lang 1987, ch. VI.3.
[55] Treves 1967, p. 34–36
[18] Roman 2005, Theorem 1.9, p. 43
[56] Lang 1983, Cor. 4.1.2, p. 69
[19] Blass 1984
[57] Treves 1967, ch. 11
[20] Halpern 1966, pp. 670–673
[58] Treves 1967, Theorem 11.2, p. 102
[21] Artin 1991, Theorem 3.3.13
[59] Evans 1998, ch. 5
[22] Braun 1993, Th. 3.4.5, p. 291
[60] Treves 1967, ch. 12
[23] Stewart 1975, Proposition 4.3, p. 52
[61] Dennery 1996, p.190
[24] Stewart 1975, Theorem 6.5, p. 74
[62] Lang 1993, Th. XIII.6, p. 349
[25] Roman 2005, ch. 2, p. 45
[63] Lang 1993, Th. III.1.1
[26] Lang 1987, ch. IV.4, Corollary, p. 106
[64] Choquet 1966, Lemma III.16.11
[27] Lang 1987, Example IV.2.6
[65] Kreyszig 1999, Chapter 11
[28] Lang 1987, ch. VI.6
[66] Griffiths 1995, Chapter 1
[29] Halmos 1974, p. 28, Ex. 9
[67] Lang 1993, ch. XVII.3
[30] Lang 1987, Theorem IV.2.1, p. 95
[68] Lang 2002, ch. III.1, p. 121
[31] Roman 2005, Th. 2.5 and 2.6, p. 49
[69] Eisenbud 1995, ch. 1.6
[32] Lang 1987, ch. V.1
[70] Varadarajan 1974
[33] Lang 1987, ch. V.3., Corollary, p. 106
[71] Lang 2002, ch. XVI.7
[34] Lang 1987, Theorem VII.9.8, p. 198
[72] Lang 2002, ch. XVI.8
[35] Roman 2005, ch. 8, p. 135–156
[73] Luenberger 1997, §7.13
[36] Lang 1987, ch. IX.4
[74] See representation theory and group representation.
[37] Roman 2005, ch. 1, p. 29
[75] Lang 1993, Ch. XI.1
[38] Roman 2005, ch. 1, p. 35
[76] Evans 1998, Th. 6.2.1
[39] Roman 2005, ch. 3, p. 64
[77] Folland 1992, p. 349 ff
[40] Lang 1987, ch. IV.3.
[78] Gasquet & Witomski 1999, p. 150
[41] Roman 2005, ch. 2, p. 48
[79] Gasquet & Witomski 1999, §4.5
[42] Mac Lane 1998
[80] Gasquet & Witomski 1999, p. 57
[43] Roman 2005, ch. 1, pp. 31–32
[81] Loomis 1953, Ch. VII
[44] Lang 2002, ch. XVI.1
[82] Ashcroft & Mermin 1976, Ch. 5
[45] Roman 2005, Th. 14.3. See also Yoneda lemma.
[83] Kreyszig 1988, p. 667
123
18.13. REFERENCES [84] Fourier 1822 [85] Gasquet & Witomski 1999, p. 67 [86] Ifeachor & Jervis 2002, pp. 3–4, 11 [87] Wallace Feb 1992 [88] Ifeachor & Jervis 2002, p. 132 [89] Gasquet & Witomski 1999, §10.2 [90] Ifeachor & Jervis 2002, pp. 307–310 [91] Gasquet & Witomski 1999, §10.3 [92] Schönhage & Strassen 1971 [93] Spivak 1999, ch. 3 [94] Jost 2005. See also Lorentzian manifold. [95] Misner, Thorne & Wheeler 1973, ch. 1.8.7, p. 222 and ch. 2.13.5, p. 325 [96] Jost 2005, ch. 3.1 [97] Varadarajan 1974, ch. 4.3, Theorem 4.3.27 [98] Kreyszig 1991, §34, p. 108 [99] Eisenberg & Guy 1979 [100] Atiyah 1989
• Lang, Serge (2002), Algebra, Graduate Texts in
Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Mac Lane, Saunders (1999), Algebra (3rd ed.), pp. 193–222, ISBN 0-8218-1646-2
• Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, SIAM, ISBN 978-0-89871-454-8
• Roman, Steven (2005), Advanced Linear Algebra, Graduate Texts in Mathematics 135 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387-24766-3
• Spindler, Karlheinz (1993), Abstract Algebra with
Applications: Volume 1: Vector spaces and groups ,
CRC, ISBN 978-0-8247-9144-5
• van der Waerden, Bartel Leendert (1993), Algebra
(in German) (9th ed.), Berlin, New York: SpringerVerlag, ISBN 978-3-540-56799-8
18.13.2 Analysis
• Bourbaki,
Nicolas (1987), Topological vector spaces, Elements of mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-13627-9
[101] Artin 1991, ch. 12
• Bourbaki, Nicolas (2004), Integration I , Berlin, New
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[103] Meyer 2000, Example 5.13.5, p. 436 [104] Meyer 2000, Exercise 5.13.15–17, p. 442 [105] Coxeter 1987
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in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer, ISBN 978-1-55608-010-4
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• Choquet, Gustave (1966), Topology, Boston, MA:
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• Dennery,
• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-89871-510-1
• Blass, Andreas (1984), “Existence of bases implies the axiom of choice”, Axiomatic set theory (Boulder, Colorado, 1983) , Contemporary Mathematics 31, Providence, R.I.: American Mathematical Society, pp. 31–33, MR 763890
• Brown, William A. (1991), Matrices and vector spaces, New York: M. Dekker, ISBN 978-0-8247-
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• Dudley,
Richard M. (1989), Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-10050-6
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Lawrence C. (1998), Partial differential equations, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0772-9
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• Folland, Gerald B. (1992), Fourier Analysis and Its Applications, Brooks-Cole, ISBN 978-0-534-
• Bourbaki, Nicolas (1969), Éléments d'histoire des
• Gasquet, Claude; Witomski, Patrick (1999), Fourier
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• Lang, Serge (1983), Real analysis, Addison-Wesley, ISBN 978-0-201-14179-5
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• Loomis, Lynn H. (1953), An introduction to abstract harmonic analysis , Toronto-New York–London: D.
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• Schaefer, Helmut H.; Wolff, M.P. (1999), Topo-
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• Fourier, Jean Baptiste Joseph (1822), Théorie ana-
lytique de la chaleur (inFrench), Chez FirminDidot,
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• Grassmann,
dehnungslehre - Ein neuer Zweig der Mathematik
(in German), O. Wigand, reprint: Hermann Grassmann. Translated by Lloyd C. Kannenberg. (2000), Kannenberg, L.C., ed., Extension Theory , Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2031-5
• Hamilton,
• Treves, François (1967), Topological vector spaces,
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• Moore, Gregory H. (1995), “The axiomatization of linear algebra: 1875–1940”, Historia Mathematica 22 (3): 262–303, doi:10.1006/hmat.1995.1025
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ondo l'Ausdehnungslehre di H. Grassmann preceduto dalle Operazioni della Logica Deduttiva (in Italian),
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logical vector spaces (2nd ed.), Berlin, New York: 18.13.4
Springer-Verlag, ISBN 978-0-387-98726-2
Hermann (1844), Die Lineale Aus-
Further references
• Ashcroft, Neil; Mermin, N. David (1976), Solid State Physics, Toronto: Thomson Learning, ISBN
978-0-03-083993-1
• Atiyah, 18.13.3
Historical references
• Banach, Stefan (1922), “Sur les opérations dans les
ensembles abstraits et leur application aux équations intégrales (On operations in abstract sets and their application to integral equations)" (PDF), Fundamenta Mathematicae (in French) 3, ISSN 0016-2736
• Bolzano, Bernard (1804), Betrachtungen über einige
Gegenstände der Elementargeometrie (Considerations of some aspects of elementary geometry) (in
German)
Michael Francis (1989), K-theory, Advanced Book Classics (2nd ed.), Addison-Wesley, ISBN 978-0-201-09394-0, MR 1043170
• Bourbaki, Nicolas (1998), Elements of Mathemat-
ics : Algebra I Chapters 1-3 , Berlin, New York:
Springer-Verlag, ISBN 978-3-540-64243-5
• Bourbaki, Nicolas (1989), General Topology. Chapters 1-4, Berlin, New York: Springer-Verlag, ISBN
978-3-540-64241-1
• Coxeter,
Harold Scott MacDonald (1987), Pro jective Geometry (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96532-1
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18.14. EXTERNAL LINKS
• Eisenberg, Murray; Guy, Robert (1979), “A proof
• Schönhage, A.; Strassen, Volker (1971), “Schnelle
• Eisenbud,
David (1995), Commutative algebra, Graduate Texts in Mathematics 150, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94269-8, MR 1322960
• Spivak, Michael (1999), A Comprehensive Introduc-
• Goldrei, Derek (1996), Classic Set Theory: A guided
Hall Mathematics Series, London: Chapman and Hall, ISBN 0-412-10800-3
of the hairy ball theorem”, The American Mathematical Monthly (Mathematical Association of America) 86 (7): 572–574, doi:10.2307/2320587, JSTOR 2320587
independent study (1st ed.), London: Chapman and
Hall, ISBN 0-412-60610-0
• Griffiths, David J. (1995), Introduction to Quantum
Mechanics , Upper Saddle River, NJ: Prentice Hall,
ISBN 0-13-124405-1
• Halmos, Paul R. (1974), Finite-dimensional vector spaces, Berlin, New York: Springer-Verlag, ISBN
978-0-387-90093-3
• Halpern, James D. (Jun 1966), “Bases in Vec-
tor Spaces and the Axiom of Choice”, Proceedings of the American Mathematical Society (American Mathematical Society) 17 (3): 670–673, doi:10.2307/2035388, JSTOR 2035388
• Husemoller, Dale (1994), Fibre Bundles (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387-94087-8
• Jost,
Jürgen (2005), Riemannian Geometry and Geometric Analysis (4th ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-25907-7
• Kreyszig, Erwin (1991), Differential geometry, New York: Dover Publications, pp. xiv+352, ISBN 9780-486-66721-8
• Kreyszig,
Erwin (1999), Advanced Engineering Mathematics (8th ed.), New York: John Wiley & Sons, ISBN 0-471-15496-2
• Luenberger, David (1997), Optimization by vector space methods , New York: John Wiley & Sons,
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Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
• Misner, Charles W.; Thorne, Kip; Wheeler, John
Archibald (1973), Gravitation, W. H. Freeman, ISBN 978-0-7167-0344-0
• Naber,
Gregory L. (2003), The geometry of Minkowski spacetime, New York: Dover Publications, ISBN 978-0-486-43235-9, MR 2044239
Multiplikation großer Zahlen (Fast multiplication of big numbers)" (PDF), Computing (in German) 7: 281–292, doi:10.1007/bf02242355, ISSN 0010485X tion to Differential Geometry (Volume Two) , Hous-
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• Stewart, Ian (1975), Galois Theory, Chapman and
• Varadarajan, V. S. (1974), Lie groups, Lie algebras, and their representations , Prentice Hall, ISBN 978-
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ture compression standard”, IEEE Transactions on Consumer Electronics 38 (1): xviii–xxxiv, doi:10.1109/30.125072, ISSN 0098-3063
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External links
• Hazewinkel, Michiel, ed.
(2001), “Vector space”,
Encyclopedia of Mathematics , Springer, ISBN 978-
1-55608-010-4
• A lecture about fundamental concepts related to vector spaces (given at MIT)
• A graphical simulator for the concepts of span, linear dependency, base and dimension
Chapter 19
Zero divisor In abstract algebra, an element a of a ring R is called a 19.1.1 One-sided zero-divisor left zero divisor if there exists a nonzero x such that ax = 0,[1] or equivalently if the map from R to R that sends x x y • Consider the ring of (formal) matrices [2] to ax is not injective. Similarly, an element aofaringis 0 z called a right zero divisor if there exists a nonzero y such with x, z ∈ Z and y ∈ Z/2Z . Then that ya = 0. This is a partial case of divisibility in rings. x y a b xa xb + yc = and An element that is a left or a right zero divisor is simply zc 0 z 0 c 0 called a zero divisor.[3] An element a that is both a left a b x y xa ya + zb = . If and a right zero divisor is called a two-sided zero divisor zc 0 c 0 z 0 (the nonzero x such that ax = 0 may be different from the ̸ 0 ≠ y , then x0 zy is a left zero divisor x = nonzero y such that ya = 0). If the ring is commutative, then the left and right zero divisors are the same. 0 1 0 x x y iff x is even, since 0 z 0 0 = 0 0 ; An element of a ring that is not a zero divisor is called and it is a right zero divisor iff z is even for similar regular, or a non-zero-divisor. A zero divisor that is reasons. If either of x, z is 0 , then it is a two-sided nonzero is called a nonzero zero divisor or a nontrivial zero-divisor. zero divisor. If there are no nontrivial zero divisors in R, then R is a division ring • Here is another example of a ring with an element that is a zero divisor on one side only. Let S be the set of all sequences of integers ( a1, a2, a3,...) 19.1 Examples . Take for the ring all additive maps from S to S , with pointwise addition and composition as the • Inthering Z/4Z , theresidueclass 2 isa zero divisor ring operations. (That is, our ring is End(S ) , since 2 × 2 = 4 = 0 . the endomorphism ring of the additive group S .) Three examples of elements of this ring are the • The only zero divisor of the ring Z of integers is 0. right shift R(a1, a2, a3,...) = ( 0, a1, a2,...) , the left shift L(a1, a2, a3,...) = (a2, a3, a4,...) • A nilpotent element of a nonzero ring is always a , and the projection map onto the first factor two-sided zero divisor. P (a1, a2, a3,...) = (a1, 0, 0,...) . All three of • An idempotent element e̸ = 1 of a ring is always a these additive maps are not zero, and the compostwo-sided zero divisor, since e (1 − e) = 0 = (1 − ites LP and P R are both zero, so L is a left zero e)e . divisor and R is a right zero divisor in the ring of additive maps from S to S . However, L is not a • Examples of zero divisors in the ring of 2 × 2 maright zero divisor and R is not a left zero divisor: the trices (over any nonzero ring) are shown here: composite LR is the identity. Note also that RL is two-sided zero-divisor since RLP = 0 = P RL , 1 1 1 1 −2 1 1 1 = 0 0 , awhile = LR = 1 is not in any direction. −2 1 2 2 2 2 −1 −1 0 0
� � � �
� �� � � � �� � � � � � �� � � �
� �� � � �� � � � � �� � � �� � � � 1 0 0 0
0 0 = 0 1
0 0 0 1
1 0
0 = 0
0 0 0 0
19.2
• A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in R1 × R 2 with each Ri nonzero, (1,0)(0,1) = (0,0), so (1,0) is a zero divisor.
126
Non-examples
• The ring of integers modulo a prime number has no zero divisors other than 0. Since every nonzero element is a unit, this ring is a field.
127
19.6. SEE ALSO
is a zero divisor on M otherwise.[4] The set of M-regular except 0. elements is a multiplicative set in R.[5] A nonzerocommutative ringwhoseonlyzerodivisor Specializing the definitions of “M-regular” and “zero divisor on M” to the case M = R recovers the definitions of is 0 is called an integral domain. “regular” and “zero divisor” given earlier in this article.
• More generally, a division ring has no zero divisors •
19.3
Properties
• In the ring of n-by-n matrices over a field, the left
and right zero divisors coincide; they are precisely the singular matrices. In the ring of n-by-n matrices over an integral domain, the zero divisors are precisely the matrices with determinant zero.
19.6 See also
• Zero-product property • Glossary of commutative algebra (Exact zero divisor)
• Left or right zero divisors can never be units,−1 be-
cause if a is invertible and ax = 0, then 0 = a 0 = 19.7 a−1ax = x , whereas x must be nonzero.
Notes
[1] See Bourbaki, p. 98.
19.4 Zero as a zero divisor There is no need for a separate convention regarding the case a = 0, because the definition applies also in this case:
• If R is a ring other than the zero ring, then 0 is a (two-sided) zero divisor, because 0 · 1 = 0 and 1 · 0 = 0.
• If R is the zero ring, in which 0 = 1, then 0 is not
a zero divisor, because there is no nonzero element that when multiplied by 0 yields 0.
Such properties areneededin order to make thefollowing general statements true:
• In a nonzero commutative ring R, the set of nonzero-divisors is a multiplicative set in R. (This, in turn, is important for the definition of the total quotient ring.) The same is true of the set of non-leftzero-divisors and the set of non-right-zero-divisors in an arbitrary ring, commutative or not.
• In a commutative Noetherian ring R, the set of zero
divisors is the union of the associated prime ideals of R.
Some references choose to exclude 0 as a zero divisor by convention, but then they must introduce exceptions in the two general statements just made.
19.5 Zero divisor on a module Let R be a commutative ring, let M be an R-module, and let a be an element of R. One says that a is M-regular if a themultiplication bya map M → M is injective, andthat a
[2] Since the map is not injective, we have ax = ay, in which x differs from y, and thus a(x-y) = 0. [3] See Lanski (2005). [4] Matsumura, p. 12 [5] Matsumura, p. 12
19.8 References
• N.
Bourbaki (1989), Algebra I, Chapters 1–3, Springer-Verlag.
• Hazewinkel, Michiel, ed.
(2001), “Zero divisor”, Encyclopedia of Mathematics , Springer, ISBN 9781-55608-010-4
• Michiel Hazewinkel; Nadiya Gubareni; Nadezhda
Mikhaĭlovna Gubareni; Vladimir V. Kirichenko. (2004), Algebras, rings and modules, Vol. 1, Springer, ISBN 1-4020-2690-0
• Charles Lanski (2005), Concepts in Abstract Algebra, American Mathematical Soc., p. 342
• Hideyuki Matsumura (1980), Commutative algebra, 2nd edition, The Benjamin/Cummings Publishing
Company, Inc.
• Weisstein, Eric W., “Zero Divisor”, MathWorld .
128
CHAPTER 19. ZERO DIVISOR
19.9
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Galois group Source: https://en.wikipedia.org/wiki/Galois_group?oldid=719899831 Contributors: AxelBoldt,
Zundark, Daniel Mahu, Michael Hardy, Dominus, TakuyaMurata, AugPi, Loren Rosen, Charles Matthews, Dysprosia, Fredrik, Giftlite, Dan Gardner, Chowbok, Vivacissamamente, EmilJ, Oleg Alexandrov, Mathbot, Dmharvey, Grubber, RDBury, Unyoyega, Lhf, Moxmalin, Keyi, Marek69, RobHar, Escarbot, JamesBWatson, Jakob.scholbach, David Eppstein, STBot, Alro, TomyDuby, Policron, Sigmundur, Hesam7, AlleborgoBot, Cwkmail, JackSchmidt, Niceguyedc, Alexbot, He7d3r, Dyaa, Legobot, Luckas-bot, AnomieBOT, MattTait, Xqbot, Point-set topologist, Howard McCay, LucienBOT, Lagelspeil, Dinamik-bot, Ebrambot, Minsbot, Zieglerk, K9re11 and Anonymous: 24
19.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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Group (mathematics) Source: https://en.wikipedia.org/wiki/Group_(mathematics)?oldid=727973745 Contributors: AxelBoldt,
Brion VIBBER, Mav, Uriyan, Bryan Derksen, Zundark, Tarquin, XJaM, Darius Bacon, Toby~enwiki, Toby Bartels, Zippy, Olivier, Patrick, Chas zzz brown, Michael Hardy, Wshun, Kku, Bcrowell, Chinju, Tango, TakuyaMurata, Jimfbleak, LittleDan, Glenn, Marco Krohn, Andres, Jonik, Revolver, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, The Anomebot, Taxman, Fibonacci, Topbanana, Raul654, Daran, PuzzletChung, Donarreiskoffer, Robbot, Gandalf61, Puckly, Rursus, Ashwin, Fuelbottle, Tea2min, Pdenapo, Tosha, Giftlite, Lee J Haywood, Michael Devore, Python eggs, Stephen Ducret, DRE, APH, Pmanderson, ELApro, Shahab, Mormegil, Rich Farmbrough, Guanabot, ArnoldReinhold, Mani1, Wadewitz, Paul August, Goochelaar, Bender235, RJHall, Joanjoc~enwiki, Hayabusa future, Art LaPella, Wood Thrush, C S, Dungodung, Helix84, Jumbuck, Msh210, Guy Harris, Sl, Hippophaë~enwiki, PAR, Woodstone, HenryLi, Oleg Alexandrov, Arneth, SP-KP, OdedSchramm, Mpatel, Jdiemer,RyanReich, Graham87, Ilya, Qwertyus,Jshadias, Chenxlee,Josh Parris, Rjwilmsi,Jarretinha, OneWeirdDude, MarSch, Pako, Salix alba, R.e.b., DoubleBlue, Penumbra2000, VKokielov, Nihiltres, Jrtayloriv, Mongreilf, Chobot, MithrandirMage, Algebraist, Debivort, YurikBot, Wavelength, Hairy Dude, Grubber, Archelon, Gaius Cornelius, Canadaduane, Rick Norwood, Dtrebbien, Kinser, PAStheLoD, DYLAN LENNON~enwiki, Natkeeran, KarlHeg, David Underdown, LarryLACa, Zzuuzz, Arthur Rubin, Redgolpe, SmackBot, Melchoir, Stifle, Gilliam, Dan Hoey, Bh3u4m, Bluebot, Soru81, Oli Filth, Silly rabbit, Nbarth, Emurphy42, Kjetil1001, Mark Wolfe, Vanished User 0001, Lesnail, TKD, LkNsngth, Nibuod, Slawekk, DMacks, Mostlyharmless, Lambiam, Harryboyles, Eriatarka, EnumaElish, Michael Kinyon, Loadmaster, Mscalculus, SandyGeorgia, Rschwieb, Markan~enwiki, Danielh~enwiki, Newone,AGK,Spindled,Paul Matthews,CRGreathouse,CmdrObot, CBM,Rawling, Myasuda, WillowW,Mike Christie, Dr.enh, Kozuch, Xantharius, Thijs!bot, Epbr123, Braveorca, Markus Pössel, Konradek, Headbomb, Paxinum, Cj67, RobHar, EdJohnston, Escarbot, Sekky, Allanhalme, JAnDbot, Ricardo sandoval, Rush Psi, East718, Magioladitis, WolfmanSF, Swpb, Ling.Nut, Jakob.scholbach, Brusegadi, SwiftBot, DAGwyn, Giggy, David Eppstein, Fbaggins, Lvwarren, Olsonist, Robin S, Pbroks13, Pomte, David Callan, IPonomarev, DrKay, RJBotting, Cspan64, Cpiral, Dispenser, Indeed123, Trumpet ma rietta 45750, Nwbeeson, Bobrek~enwiki, Ginpasu, Treisijs, OktayD, LokiClock, TheOtherJesse, Philip Trueman, GimmeBot, JasonASmith, Nxavar, Anonymous Dissident, VictorMak, Skylarkmichelle, Geometry guy, Eubulides, BigDunc, Synthebot, Pjoef, AlleborgoBot, Teresol, Drschawrz, SieBot, Calliopejen1, YonaBot, Gerakibot, Soler97, Antzervos, Kareekacha, Thehotelambush, JackSchmidt, Skippydo, Jorgen W, Anchor Link Bot, S2000magician, Randomblue, CBM2, Peiresc~enwiki, A legend, Felizdenovo, Amahoney, Nergaal, Classicalecon, ClueBot, Alksentrs, Nsk92, JP.Martin-Flatin, Piledhigheranddeeper, Eeekster, Brews ohare, Cenarium, Jotterbot, Hans Adler, Wikidsp, Thingg, Dank, Qwfp, Johnuniq, TimothyRias, Basploeger, Marc van Leeuwen, Alecobbe, Kakila, GabeAB, Porphyro, CàlculIntegral, Addbot, DOI bot, Delaszk, LinkFA-Bot, Ozob, Ettrig, Luckas-bot, Yobot, WikiDan61, TaBOT-zerem, Pcap, AnomieBOT, WinoWeritas, Jarmiz, Citation bot, Frankenpuppy, Xqbot, Farvin111, X Pacman X, X Fallout X, Mee26, AYSH AYSH AY AY AY AY, Isheden, Point-set topologist, RibotBOT, Charvest, Harry007754, FrescoBot, Citation bot 1, HRoestBot, Wikitanvir, Jujutacular, RjwilmsiBot, Jowa fan, EmausBot, M759, Slawekb, ZéroBot, Josve05a, Quondum, D.Lazard, Git2010, Wayne Slam, Mentibot, ChuispastonBot, ClueBot NG, IfYouDoIfYouDon't, Tideflat, Frietjes, Mesoderm, BTotaro, Widr, Bibcode Bot, Brad7777, Nadapez~enwiki, ChrisGualtieri, Dexbot, Mark L MacDonald, Jochen Burghardt, Mark viking, CsDix, Itc editor2, Blackbombchu, Schwatzwutz, Gianluca.baldassarre, Khuramawais, UY Scuti, Anrnusna, Sansam131192, Monkbot, Levi12349, MissouriOzark1947, IPalpedia, Y2N1-09631, Mrossen and Anonymous: 219
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Group theory Source: https://en.wikipedia.org/wiki/Group_theory?oldid=722339834 Contributors: AxelBoldt,
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Homomorphism Source: https://en.wikipedia.org/wiki/Homomorphism?oldid=699443243 Contributors: AxelBoldt,BryanDerksen, Zun-
Zundark, The Anome, KF, Cwitty, Edward, Michael Hardy, Wshun, Dcljr, TakuyaMurata, Ellywa, JWSchmidt, Bogdangiusca, Poor Yorick, Rossami, Jordi Burguet Castell, Charles Matthews, Lfh, Dysprosia, Jitse Niesen, Hyacinth, Fibonacci, Phys, Bevo, Kwantus, Finlay McWalter, PuzzletChung, Gromlakh, Romanm, Mayooranathan, Gandalf61, MathMartin, Rursus, Papadopc, ComplexZeta, Tea2min, Giftlite, Graeme Bartlett, Recentchanges, Dratman, Doshell, LiDaobing, Alberto da Calvairate~enwiki, Karl-Henner, Rich Farmbrough, FT2, Luqui, ArnoldReinhold, H00kwurm, Paul August, Tompw, Jaimedv, Adan, Obradovic Goran, Friviere, Ranveig, Masv~enwiki, HenryLi, Oleg Alexandrov, Tbsmith, Archie Paulson, OdedSchramm, Kmg90, PeterPearson, DaveApter, V8rik, BD2412, Chun-hian, Josh Parris, Rjwilmsi, Dennis Estenson II, Salix alba, Ligulem, R.e.b., Brighterorange, FlaBot, Chris Pressey, Mathbot, Margosbot~enwiki, Rune.welsh, MTC, Chobot, YurikBot, Hairy Dude, Hillman, Michael Slone, Grubber, Cate, Merlincooper, Petter Strandmark, DYLAN LENNON~enwiki, Crasshopper, Googl, Tigershrike, Willtron, GrinBot~enwiki, RonnieBrown, Palapa, SmackBot, Reedy, Melchoir, Scullin, Natebarney, Cessator, BiT, GBL, Bluebot, Pieter Kuiper, MalafayaBot, Ligulembot, Pilotguy, Davipo, Christopherodonovan, Lambiam, Richard L. Peterson, Utopianheaven, Mike Fikes, Tawkerbot2, Chetvorno, CRGreathouse, Ale jrb, Gregbard, Rifleman 82, Tyskis, Mungomba, Headbomb, WVhybrid, Nadav1, RobHar, NERIUM, Escarbot, Seaphoto, M cuffa, VictorAnyakin, JAnDbot, The Transhumanist, Bongwarrior, Jakob.scholbach, CountingPine, Baccyak4H, Gabriel Kielland, David Eppstein, MaEr, R'n'B, David Callan, J.delanoy, Cmbankester, Indeed123, Gombang, Treisijs, Useight, Lemonaftertaste, VolkovBot, JohnBlackburne, EchoBravo, Philip Trueman, Eakirkman, Magmi, Eubulides, ArzelaAscoli, Arcfrk, Andreas Carter, GoddersUK, Peter Stalin, Drschawrz, SieBot, Ivan Štambuk, WereSpielChequers, Viskonsas, Messagetolove, Lightmouse, JackSchmidt, NobillyT, StaticGull, Alpha Beta Epsilon, Justin W Smith, Alksentrs, Padicgroup, Bhuna71, Mspraveen, Avouac, Watchduck, Edwinconnell, Xylthixlm, Hans Adler, Vegetator, Johnuniq, TimothyRias, XLinkBot, JinJian, CàlculIntegral, Addbot, Manuel Trujillo Berges, SpellingBot, Fluffernutter, Kristine8~enwiki, Favonian, Tide rolls, Luckas-bot, Yobot, TaBOT-zerem, Julia W, Eamonster, AnomieBOT, DemocraticLuntz, Rubinbot, Μυρμηγκάκι, WinoWeritas, Citation bot, Calcio33, Auclairde, FrescoBot, Lothar von Richthofen, Orhanghazi, Sławomir Biały, Citation bot 1, Boulaur, Hard Sin, Hamtechperson, Ngyikp, D stankov, Jauhienij, Debator of mathematics, Lightlowemon, Orenburg1, FoxBot, Yger, SomeRandomPerson23, EmausBot, Fly by Night, Tommy2010, Shishir332, D15724C710N,Quondum, Kranix, Wikiloop, Adgjdghjdety, Gottlob Gödel, ClueBot NG, LordRoem, Ciro.santilli, HMSSolent, BG19bot, Ijgt, CimanyD, Meclee, Brad7777, Jochen Burghardt, Brirush, CsDix, Laxfan1977, Chetan bagora, Edmundthe, KasparBot, Kubbiebeef and Anonymous: 143 dark, The Anome, XJaM, Rgamble, Toby Bartels, Youandme, Michael Hardy, TakuyaMurata, Ellywa, Glenn, AugPi, Netsnipe, Andres, Dysprosia, Zero0000, Ed g2s, Bloodshedder, Robbot, Gandalf61, Hadal, Tea2min, Giftlite, Fropuff, Paul August, Goochelaar, El C, Rgdboer, EmilJ, Sebastian Goll, Msh210, Mlm42, Dirac1933, Oleg Alexandrov, Robert K S, Graham87, Salix alba, Oblivious, Ligulem, YurikBot, Michael Slone, Rat144, Tong~enwiki, Bota47, KnightRider~enwiki, SmackBot, Jyoshimi, Mgreenbe, JAn Dudík, Bluebot, Riteshsood, Korako, Kevinbsmith, SashatoBot, Jim.belk, CRGreathouse, Wowulu, Hukkinen, Kanags, Vanished User jdksfajlasd, Vantelimus, Allanhalme, JAnDbot, Magioladitis, EagleFan, David Eppstein, Martynas Patasius, JoergenB, Electiontechnology, Kostisl, R'n'B, TheSeven, LokiClock, Ishboyfay, ToePeu.bot, Sahar Tomer, Pramcom, Classicalecon, WikiBotas, FractalFusion, Qwfp, Addbot, LaaknorBot, Wikomidia, PV=nRT, Yobot, Pcap, Calle, AnomieBOT, Nishantjr, Clément Pillias, Ex13, FrescoBot, Wassermann7, EmausBot, John of Reading, Quondum, D.Lazard, ChuispastonBot, Mehdi.manshadi, FeatherPluma, Anita5192, Mesoderm, Helpful Pixie Bot, BG19bot, Beaumont877, Johntyree, Deltahedron, Jochen Burghardt, Café Bene, JMP EAX, Kiwifist, Rashteh, Chemistry1111 and Anonymous: 70
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Ideal (ring theory) Source: https://en.wikipedia.org/wiki/Ideal_(ring_theory)?oldid=724404453 Contributors: AxelBoldt, Bryan Derksen,
Zundark, Taral, Gianfranco, Toby Bartels, Youandme, Ram-Man, Patrick, Michael Hardy, Chris-martin, TakuyaMurata, Stevan White, Snoyes, Ideyal, Charles Matthews, Dysprosia, Jitse Niesen, Rik Bos, Jni, Aleph4, Robbot, Mattblack82, MathMartin, Tea2min, Tosha,
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CHAPTER 19. ZERO DIVISOR Giftlite, Gene Ward Smith, Markus Krötzsch, Lupin, Dratman, Ssd, Waltpohl, Jorge Stolfi, Profvk, Gauss, Grunt, Noisy, Nparikh, Guanabot, Paul August, Elwikipedista~enwiki, Rgdboer, Haham hanuka, Mdd, LutzL, Jérôme, Arthena, Keenan Pepper, Oleg Alexandrov, Graham87, Staecker, Chobot, YurikBot, Dmharvey, Grubber, Archelon, Pmdboi, -oo-~enwiki, Ms2ger, Arthur Rubin, Netrapt, Pred, Prodego, Eskimbot, Dan Hoey, Bluebot, JCSantos, Bird of paradox, Lubos, Cícero, Zdenes, Mets501, Ravi12346, Rschwieb, Catherineyronwode, Zero sharp, CRGreathouse, CmdrObot, Mon4, Thijs!bot, RobHar, JAnDbot, GromXXVII, Kprateek88, Magioladitis, Reminiscenza, David Eppstein, Maurice Carbonaro, POP JAM, Jnlin, VolkovBot, Joeldl, GirasoleDE, Henry Delforn (old), PerryTachett, Functor salad, ClueBot, He7d3r, PixelBot, BOTarate, Beroal, Marc van Leeuwen, SilvonenBot, D.M. from Ukraine, Addbot, Lightbot, Luckasbot, KamikazeBot, ArthurBot, Xqbot, GrouchoBot, Charvest, FrescoBot, Citation bot 1, RedBot, DreamingInRed~enwiki, Dinamik-bot, Emakarov, EmausBot, John of Reading, WikitanvirBot, JasonSaulG, Josve05a, Quondum, Zeugding, Helpful Pixie Bot, Manoguru, Mdsalim786, Mcshantz, Mathedu, SoSivr and Anonymous: 92 Integral domain Source: https://en.wikipedia.org/wiki/Integral_domain?oldid=716546383 Contributors: AxelBoldt, Bryan Derksen, Zundark, XJaM, Michael Hardy, TakuyaMurata, GTBacchus, Karada, Ciphergoth, Caramdir~enwiki, Clausen, Loren Rosen, Dysprosia, MathMartin, Henrygb, Giftlite, Fropuff, Dratman, Waltpohl, Jason Quinn, Pmanderson, Elroch, Sam Hocevar, Rpchase, Barnaby dawson, Vivacissamamente, Abar, Gauge, El C, EmilJ, Jumbuck, Eric Kvaalen, Arthena, Oleg Alexandrov, Tbsmith, Joriki, Arneth, Miaow Miaow, Hypercube~enwiki, MFH, Chenxlee, Mike Segal, FlaBot, VKokielov, YurikBot, Dmharvey, Philopedia, Arthur Rubin, KnightRider~enwiki, SmackBot, Selfworm, InverseHypercube, SGNDave, Mhss, Chris the speller, SMP, BlackFingolfin, Ninte, Physis, Mets501, Rschwieb, CRGreathouse, CmdrObot, Xantharius, RobHar, Salgueiro~enwiki, JAnDbot, Vanish2, JamesBWatson, Falcor84, Lucky Eight, Squids and Chips, VolkovBot, Fukiishi, Reddyuday, Arcfrk, SieBot, ToePeu.bot, Gherrington, JackSchmidt, Anchor Link Bot, Ζητεω, JP.Martin-Flatin, Cello3141, He7d3r, Bender2k14, Marc van Leeuwen, Addbot, Expz, DOI bot, LaaknorBot, Ozob, Matěj Grabovský, Legobot, Yobot, TaBOT-zerem, Calle, AnomieBOT, Galoubet, Citation bot, Xqbot, Depassp, Druiffic, FrescoBot, LucienBOT, Artem M. Pelenitsyn, Sławomir Biały, Ebony Jackson, Modamoda, Foobarnix, Brambleclawx, H.ehsaan, WikitanvirBot, Fly by Night, Quondum, D.Lazard, L.tian.wiki, Helpful Pixie Bot, MerryTaliban1, AvocatoBot, Solomon7968, CitationCleanerBot, Deltahedron, Greatuser, Mathedu, CsDix, Fylgia Fock, Cvchaparro, Magmalord, Gjbayes, GeoffreyT2000, Arghya Chakraborty (Mathematician) and Anonymous: 60 Isometry Source: https://en.wikipedia.org/wiki/Isometry?oldid=727016498 Contributors: Zundark, Patrick, Michael Hardy, TakuyaMurata, Glenn, Charles Matthews, KRS, Hyacinth, Alembert~enwiki, Robbot, MathMartin, Lupo, Tosha, Giftlite, Lupin, Fropuff, Mike Rosoft, Reuben, Snowolf, Jopxton, Oleg Alexandrov, Peya, Isnow, Marudubshinki, Grammarbot, Salix alba, FlaBot, [email protected], Mathbot, Siddhant, YurikBot, RussBot, Gaius Cornelius, Crasshopper, ManoaChild, Bota47, Silly rabbit, Jtabbsvt, UKER, LaMenta3, Jackzhp, Sniffnoy, TheTito, Thijs!bot, B-80, Turgidson, Vanish2, Albmont, Trioculite, Sullivan.t.j, JoergenB, TomyDuby, Trumpet marietta 45750, VolkovBot, Matematico~enwiki, SieBot, Harry-, Marino-slo, Niceguyedc, DragonBot, Addbot, AkhtaBot, Topology Expert, TutterMouse, Uscitizenjason, Peti610botH, KamikazeBot, 4th-otaku, AnomieBOT, Omnipaedista, Point-set topologist, RibotBOT, EmausBot, WikitanvirBot, Quondum, Aughost, Mfluch, ClueBot NG, Wcherowi, BG19bot, Vagobot, Brad7777, Brirush, Limittheorem, Noix07, Mr. Smart LION, GeoffreyT2000, Gblikas, Hmcaun and Anonymous: 53 Magma (algebra) Source: https://en.wikipedia.org/wiki/Magma_(algebra)?oldid=722061698 Contributors: AxelBoldt, Zundark, Toby Bartels, Qwitchibo, SGBailey, AugPi, Rmilson, Andres, Charles Matthews, Doradus, 1984, Romanm, P0lyglut, Tea2min, Giftlite, Lethe, Fropuff, Jason Quinn, Jpp, Gubbubu, DefLog~enwiki, Pgan002, Phe, Habitue, EmilJ, Kevin Lamoreau, SpeedyGonsales, Linas, Shreevatsa, Uncle G, Mpatel, Wbeek, Chenxlee, Josh Parris, Rjwilmsi, Salix alba, FlaBot, Nihiltres, YurikBot, Hairy Dude, Dmharvey, Archelon, Bruguiea, Bota47, Reyk, Tropylium, GrinBot~enwiki, RupertMillard, SmackBot, Melchoir, PJTraill, Bluebot, RDBrown, Nbarth, J. Finkelstein, Michael Kinyon, Rschwieb, CRGreathouse, Myasuda, Cydebot, Thijs!bot, Kilva, Lovibond, Salgueiro~enwiki, Deom, Jakob.scholbach, Hey Ho Let’s Dave, Sullivan.t.j, David Eppstein, LokiClock, Anonymous Dissident, Popopp, Cnilep, Ponyo, YohanN7, Hawk777, Markus Prokott, Thehotelambush, JackSchmidt, Stfg, Mr. Granger, DavidHobby, Mathemajor, SamuelTheGhost, MystBot, Addbot, Tjlaxs, Ersik, AgadaUrbanit, ., Legobot, Luckas-bot, Yobot, Nallimbot, AnomieBOT, Xqbot, Nishantjr, Charvest, Serols, ElNuevoEinstein, Xnn, EmausBot, Jimj316, Quondum, BG19bot, Solomon7968, Brad7777, Papxr, Nadapez~enwiki, IkamusumeFan, Deltahedron, Natuur12, Mark viking, CsDix, JP.Martin-Flatin (old), Gcleaner66553377, JMP EAX, M80126colin, Anareth and Anonymous: 31 Order (group theory) Source: https://en.wikipedia.org/wiki/Order_(group_theory)?oldid=709205934 Contributors: AxelBoldt, TakuyaMurata, Charles Matthews, Robbot, Schutz, Lowellian, Tea2min, Giftlite, BenFrantzDale, Waltpohl, Dnas, Tomruen, Pyrop, Gauge, Ntmatter, Hoziron, Gadlor, VKokielov, YurikBot, RussBot, KSmrq, Gwaihir, Voidxor, Chris the speller, AdamSmithee, JRSpriggs, CRGreathouse, Thijs!bot, Kilva, F.mardini,Marek69, Albmont, JoergenB, Yensin, TomyDuby, Skylarkmichelle,Mrinsuperable, JackSchmidt, Andrewbt, Razimantv, He7d3r, Beroal, Addbot, Luckas-bot, Yobot, AnomieBOT, , FoxBot, Quondum, D.Lazard, ChuispastonBot, Kasirbot, Walk&check, Cnorrisl, SoSivr, ArcanaNoir and Anonymous: 20 Ring (mathematics) Source: https://en.wikipedia.org/wiki/Ring_(mathematics)?oldid=727126347 Contributors: Damian Yerrick, AxelBoldt, Bryan Derksen, Zundark, Tarquin, Youssefsan, Toby Bartels, Miguel~enwiki, Patrick, Michael Hardy, Wshun, Dominus, TakuyaMurata, Stevan White, Stan Shebs, Suisui, Angela, AugPi, Rotem Dan, Andres, Clausen, Vargenau, Schneelocke, Ideyal, Ffransoo, Loren Rosen, Revolver, Charles Matthews, Dysprosia, Jitse Niesen, Kbk, Prumpf, Itai, Taxman, VeryVerily, Aleph4, Robbot, Romanm, MathMartin, Henrygb, Puckly, Bkell, Tea2min, Tosha, Giftlite, BenFrantzDale, Tom harrison, Fropuff, Berjoh, FunnyMan3595, Michael Devore, Jorend, Ssd, Jorge Stolfi, Python eggs, Chowbok, Sigfpe, Profvk, Pmanderson, Mschlindwein, Frenchwhale, Barnaby dawson, Vivacissamamente, PhotoBox, D6, Smimram, Discospinster, Hydrox, Mecanismo, Xezbeth, Paul August, Gauge, Syp, [email protected], El C, Rgdboer, EmilJ, O18, Touriste, La goutte de pluie, Nk, Obradovic Goran, Msh210, Eric Kvaalen, AidanH, Emvee~enwiki, Oleg Alexandrov, Joriki, Linas, Aaron McDaid, Robert K S, Halcatalyst, Mandarax, Graham87, Ilya, Jclemens, Chenxlee, Radomir, Rjwilmsi, OneWeirdDude, Pako, Staecker, Salix alba, Mathbot, R160K, Sunil.nandihalli, Chobot, Bgwhite, Algebraist, Jayme, Wavelength, Hairy Dude, Dmharvey, Michael Slone, Pi Delport, KSmrq, Grubber, Chrispounds, ENeville, Rick Norwood, NickBush24, David Pierce, Crasshopper, Pooryorick~enwiki, 2over0, Arthur Rubin, Netrapt, Pred, Bo Jacoby, Jsnx, SmackBot, Selfworm, InverseHypercube, CapitalSasha, Cazort, Hmains, Anastasios~enwiki, JasonMR, Snori, PrimeHunter, MalafayaBot, Silly rabbit, Octahedron80, Don Hosek, AdamSmithee, Kjetil1001, Cybercobra, Diocles, Tilin, Ninte, Soumyasch, Rschwieb, Ojan, CRGreathouse, DavidFHoughton, Nightwriter50, Ironmagma, Myasuda, Mct mht, Marqueed, Dl573, Gogo Dodo, Goldencako, Xantharius, Thijs!bot, Wikid77, Andri Egilsson, Headbomb, RobHar, OrenBochman, CZeke, Papipaul, JAnDbot, Drizzd~enwiki, Magioladitis, JamesBWatson, Swpb, Jakob.scholbach, Alvian, Twsx, WhatamIdoing, JJ Harrison, David Eppstein, JaGa, Tejon~enwiki, Pomte, Laurusnobilis, TomyDuby, Jeepday, OliverHarris, Yecril, VolkovBot, JohnBlackburne, LokiClock, AlnoktaBOT, Cbigorgne, TXiKiBoT, Malsaqer, Rei-bot, Anonymous Dissident, Marcosaedro, Plclark, Gillyrules18, Joeldl, Bphenry, AlleborgoBot, Ocsenave, SieBot, Ivan Štambuk, X-Fi6, Wing gundam, Yerpo, Henry Delforn (old), Thehotelambush, JackSchmidt, Deadlyhair, Int21h, Jorgen W, OKBot, Oekaki, The Thing That Should Not Be, JP.Martin-Flatin, Mild Bill Hiccup, Niceguyedc, Jalanpalmer, NClement, He7d3r, Hans Adler, Sterlesser, Jshen6, Lambtron, WikHead, Addbot, Roentgenium111,
19.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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Expz, AkhtaBot, Zorrobot, Legobot, Luckas-bot, Yobot, AnomieBOT, Law, Citation bot, Jellystones, LilHelpa, Xqbot, Andreesakia, GeometryGirl, Nfr-Maat, Omnipaedista, VladimirReshetnikov, Point-set topologist, Charvest, Raulshc, FrescoBot, Sławomir Biały, Citation bot 1, Jlaire, Velvetfish, Redrose64, Ebony Jackson, HRoestBot, Sa'y, MastiBot, Wallacoloo, Foobarnix, Beao, Im not afraid, Trappist the monk, Lovek323, EmausBot, GoingBatty, KHamsun, Slawekb, Adolfbatman, Quondum, D.Lazard, Wikfr, ChuispastonBot, RockMagnetist, Ebehn, Nitrogl, Anita5192, ClueBot NG, El Roih, Matthiaspaul, Derschmidt, Frietjes, MerlIwBot, Helpful Pixie Bot, BG19bot, John Cummings, David815, QualitativeMath, CitationCleanerBot, Dlituiev, MohaiminPatwary, ChrisGualtieri, Khazar2, Deltahedron, Zeeyanwiki, Mark L MacDonald, Remag12, Mark viking, Lobote, AcheronSS, Mathedu, CsDix, Purnendu Karmakar, Barrettcw, Critical Reason, Craed, Rawatsanjay, Tarpuq, Magmalord, Lilyemmab, Kirstyoulden, Revstifeev, Peiffers, Unician, Fyodorjung, Jmcgnh and Anonymous: 191 Subgroup Source: https://en.wikipedia.org/wiki/Subgroup?oldid=712011049 Contributors: AxelBoldt, Zundark, Patrick, Chas zzz brown, Michael Hardy, TakuyaMurata, AugPi, Revolver, Charles Matthews, Dcoetzee, Dysprosia, Zero0000, Fredrik, Schutz, Jotomicron, MathMartin, Giftlite, DocWatson42, Arved, Lethe, Simplex~enwiki, Pyrop, Adan, Vipul, Viriditas, Obradovic Goran, Storm Rider, Oleg Alexandrov, SLi, VKokielov, Mathbot, Margosbot~enwiki, Bgwhite, Roboto de Ajvol, YurikBot, RussBot, Michael Slone, Grubber, Gaius Cornelius, Zwobot, Bota47, Pred, SmackBot, Edgar181, Nbarth, Jim.belk, Mets501, Noleander, Wakimakirolls, Thijs!bot, Kilva, Magioladitis, Jakob.scholbach, David Eppstein, Arturj, OktayD, LokiClock, Rei-bot, Quietbritishjim, Anchor Link Bot, Razimantv, Watchduck, He7d3r, Jaan Vajakas, Addbot, Topology Expert, West.andrew.g, Zorrobot, Legobot, Luckas-bot, Yobot, JackieBot, ArthurBot, Xqbot, Point-set topologist, HRoestBot, Jonesey95, RedBot, Tanner Swett, ZéroBot, Quondum, Jadzia2341, Augurar, MRG90, Jeremy112233, CsDix, K9re11, SoSivr and Anonymous: 28 Symmetry Source: https://en.wikipedia.org/wiki/Symmetry?oldid=729003485 Contributors: AxelBoldt, Bryan Derksen, The Anome, Tarquin, XJaM, Stevertigo, Patrick, Michael Hardy, Earth, HarmonicSphere, Haakon, CatherineMunro, Cyan, AugPi, Marksuppes, Rossami, Jeandré du Toit, Smack, RodC, Charles Matthews, Jitse Niesen, Wik, Zoicon5, Maximus Rex, Jeffrey Smith, Hyacinth, Nv8200pa, Phys, Omegatron, Bevo, Olathe, Finlay McWalter, Jeffq, Robbot, Altenmann, Romanm, Mayooranathan, Gandalf61, Rursus, Burtonator, JeffC, Paul Murray, ElBenevolente, Alanyst, Tea2min, Marc Venot, Giftlite, Pandammonium, Tom harrison, Herbee, Dratman, DO'Neil, Gracefool, Andycjp, Geni, Yarnover, Quadell, Antandrus, Joseph Myers, Tomruen, C4~enwiki, Porges, Venu62, Imroy, Discospinster, Paul August, Bender235, Tompw, Rgdboer, Hayabusa future, Uieoa, Mpulier, Benbread, Alansohn, BadSanta, CuriousOne, Jheald, Jon Cates, Oleg Alexandrov, Feezo, Stemonitis, Mel Etitis, Woohookitty, Linas, David Haslam, Guy M, Ruud Koot, Tabletop, GregorB, Pfalstad, Marudubshinki, Graham87, Magister Mathematicae, V8rik, BD2412, Martin von Gagern, Pmj, KYPark, Stardust8212, Salix alba, SchuminWeb, Mathbot, Margosbot~enwiki, Nihiltres, RexNL, Ben Babcock, SpectrumDT, King of Hearts, Scoops, Eric B, PointedEars, YurikBot, Wavelength, RussBot, Splash, Grubber, CambridgeBayWeather, NawlinWiki, 0waldo, Wiki alf, Anetode, Mysid, Trojanavenger, Tomabbott, Enormousdude, Cullinane, Arthur Rubin, MathsIsFun, Willtron, Eugcc, GrinBot~enwiki, MelRip, Segv11, AceVentura, Sbyrnes321, RonnieBrown, Brentt, Lviatour, SmackBot, Incnis Mrsi, Melchoir, Pavlovič, Pgk, KocjoBot~enwiki, Delldot, Atomota, Xaosflux, Gilliam, Ennorehling, Bluebot, Fplay, MalafayaBot, Silly rabbit, CSWarren, Ikiroid, Octahedron80, Kostmo, John Reaves, Antabus, Tamfang, OrphanBot, Rrburke, Cybercobra, TheLimbicOne, Akriasas, Sadi Carnot, ArglebargleIV, Ybact, Lakinekaki, Terry Bollinger, Bjankuloski06en~enwiki, Firefox13, 16@r, Violncello, Abel Cavaşi, Dreftymac, MIckStephenson, Joseph Solis in Australia, Theone00, S0me l0ser, 'Ff'lo, Debanjum, JForget, CmdrObot, Iced Kola, The Font, CBM, MarsRover, Shadow12l, Cydebot, Rifleman 82, JFreeman, Jgbeldock, Xminivann, Vanished User jdksfajlasd, Thijs!bot, Epbr123, Kilva, Mojo Hand, Miesling, Yaragn, TimVickers, Coyets, Darvasg, Steelpillow, JAnDbot, Txomin, Struthious Bandersnatch, Dcooper, Bongwarrior, Dekimasu, Soulbot, Baccyak4H, SparrowsWing, Johnbibby, Seberle, Justanother, Japo, David Eppstein, Cpl Syx, DerHexer, JaGa, Wdflake, Khalid Mahmood, Pax:Vobiscum, Falcor84, Monurkar~enwiki, Gwern, Jtir, MartinBot, Schmloof, R'n'B, Pomte, Wlodzimierz, J.delanoy, BigrTex, Nigholith, Chiswick Chap, Lbeaumont, Vanished user 39948282, DWPittelli, VolkovBot, Thisisborin9, JohnBlackburne, Soliloquial, AllS33ingI, Philip Trueman, TXiKiBoT, Bbik, CosmonautLaunchPad, Mercurywoodrose, Zamphuor, A4bot,Weena Eloi, JohnEllsworth, LeaveSleaves, Falcon8765, Brianga, Hrafn, SieBot, Nubiatech, Malcolmxl5, Caltas, Keilana, Bentogoa, Flyer22 Reborn, Oxymoron83, JerroldPease-Atlanta, JackSchmidt, Onopearls, Termer, Nn123645, Denisarona, ImageRemovalBot, Tanvir Ahmmed, ClueBot, Snigbrook, The Thing That Should Not Be, BenWillard, Hal8999, Hafspajen, DragonBot, Watchduck, Ottre, Sun Creator, Brews ohare, Promethean, Hans Adler, SchreiberBike, Taranet, Vybr8, Qwfp, TimothyRias, XLinkBot, Dthomsen8, Petitjeanmichel, HOTmag, Addbot, Jpiñacheeto, MrOllie, 5 albert square, Xev lexx, PjOfAustralia, Wytenus208, Tide rolls, Zorrobot, Snaily, Legobot, Cote d'Azur, Luckas-bot, Yobot, Pink!Teen, NotARusski, Gobbleswoggler, AnomieBOT, Jim1138, Materialscientist, Citation bot, ArthurBot, Xqbot, Joshua.mccall, The Evil IP address, GrouchoBot, Lillebi, Omnipaedista, RibotBOT, Shadowjams, Hersfold tool account, FrescoBot, Finalius, Pinethicket, Thinking of England, RobinK, TobeBot, Imaebn, Lotje, Vancouver Outlaw, Jave7784, Bea.miau, Onel5969, TjBot, DexDor, John of Reading, Kpufferfish, RA0808, ZxxZxxZ, Dcirovic, ZéroBot, PBS-AWB, Fæ, Bollyjeff, Parsonscat, Aknicholas, Wayne Slam, Ben Tamari, Jacobisq, JaySebastos, Donner60, Scientific29, ChuispastonBot, Weimer, ClueBot NG, Malleus Felonius, Scalelore, Graythos1, Widr, , Darian25, Helpful Pixie Bot, Bibcode Bot, BG19bot, Snaevar-bot, Qx2020, Канеюку, Cyberpower678, MusikAnimal, Krupasindhu Muduli, EditorRob, Klilidiplomus, Wannabemodel, Huntlj88, David.moreno72, Tonusamuel, Harshul Ravindran, GoShow, Khazar2, EuroCarGT, Kelvinsong, Volvens, Mogism, Brirush, Leprof 7272, Mark viking, Tentinator, Oj.jain, Ginsuloft, Argent2, Monkbot, Yikkayaya, Crystallizedcarbon, Rakshith12Kiran, Loraof, Malc9141, Yolo pizza pocket, Redzemp, New User Person, Krishikaran, HannuMannu, TheGoldenParadox, Lily puff and Anonymous: 391 Symmetry group Source: https://en.wikipedia.org/wiki/Symmetry_group?oldid=718983717 Contributors: AxelBoldt, Tarquin, Jkominek, JoshGrosse, Nonenmac,Stevertigo, Edward, Patrick, Dominus,Stevenj, Charles Matthews,Dysprosia, AndrewKepert, RedWolf,Sverdrup, Tea2min, Giftlite, Snags, BenFrantzDale, Fropuff, Jason Quinn, Auximines, Beland, Bornintheguz, Rich Farmbrough, Qutezuce, Fadereu, Oleg Alexandrov, MFH, Tokek, BD2412, Martin von Gagern, Eubot, Mathbot, Nihiltres, Debivort, Siddhant, YurikBot, Wavelength, Reverendgraham, KSmrq, Raven4x4x, LarryLACa, Cullinane, Modify, Wikipedist~enwiki, IstvanWolf, TimBentley, Silly rabbit, Tsca.bot, Tamfang, TheLimbicOne, Pcgomes, Vina-iwbot~enwiki, Maverick starstrider, Mets501, Sir Vicious, Thijs!bot, Kilva, Hannes Eder, Dirac66, R'n'B, JohnBlackburne, LokiClock, TXiKiBoT, Anonymous Dissident, Eubulides, Anchor Link Bot, Addbot, Romaioi, Luckasbot, Xqbot, GrouchoBot, Lillebi, Sławomir Biały, RobinK, EmausBot, Minimac’s Clone, Slawekb, Quondum, Jadzia2341, Maschen, DonBex, ClueBot NG, Helpful Pixie Bot, Russell157 and Anonymous: 30 Vector field Source: https://en.wikipedia.org/wiki/Vector_field?oldid=720274140 Contributors: AxelBoldt, Chato, Patrick, Chas zzz brown, Michael Hardy, Tim Starling, Wshun, TakuyaMurata, Cyp, Stevenj, Andres, Charles Matthews, Reddi, Sbwoodside, Dysprosia, Jitse Niesen, MaximusRex, Fibonacci, Phys, Jaredwf, MathMartin, Idoneus~enwiki, Tosha, Giftlite, BenFrantzDale, Ævar Arnfjörð Bjarmason, DefLog~enwiki, LiDaobing, MFNickster, Hellisp, JohnArmagh, Zowie, Klaas van Aarsen, Rich Farmbrough, ReiVaX, Bender235, Mdd, Wendell, Oleg Alexandrov, Woohookitty, Linas, Jacobolus, Rjwilmsi, MarSch, HannsEwald, Salix alba, Dergrosse, Mo-Al, FlaBot, Margosbot~enwiki, Alfred Centauri, Chobot, 121a0012, WriterHound, YurikBot, Archelon, Buster79, Mgnbar, Darrel francis, Sbyrnes321, SmackBot, RDBury, Rex the first, Pokipsy76, Silly rabbit, Nbarth, DHN-bot~enwiki, Regford, Daqu, Pen of bushido, Andrei Stroe, Cron-
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CHAPTER 19. ZERO DIVISOR holm144, Jim.belk, Dwmalone, FelisSchrödingeris, Thijs!bot, KlausN~enwiki, JAnDbot, Rivertorch, Catgut, Sullivan.t.j, User A1, Martynas Patasius, JaGa, Rickard Vogelberg, R'n'B, TomyDuby, Policron, HyDeckar, Jaimeastorga2000, VolkovBot, LokiClock, Julian I Do Stuff, TXiKiBoT, A4bot, Anonymous Dissident, Michael H 34, Geometry guy, Antixt, SieBot, Soler97, Paolo.dL, JackSchmidt, OKBot, 7&6=thirteen, Wikidsp, Addbot, Fgnievinski, Topology Expert, EconoPhysicist, AndersBot, Mattmatt79, Jasper Deng, Zorrobot, Jarble, Snaily, Legobot, Luckas-bot, Yobot, Naudefjbot~enwiki, AnomieBOT, Ciphers, ArthurBot, Titi2~enwiki, Point-set topologist, FrescoBot, Lookang, Sławomir Biały, Lost-n-translation, Foobarnix, Tcnuk, Rausch, EmausBot, Fly by Night, Slawekb, Hhhippo, ZéroBot, Qniemiec, Wikfr, Glosser.ca, Sp4cetiger, Helpsome, Wcherowi, Clearlyfakeusername, Snotbot, Frietjes, Mesoderm, Vinícius Machado Vogt, Helpful Pixie Bot, Shivsagardharam, Nawk, MusikAnimal, Cispyre, F=q(E+v^B), ChrisGualtieri, Svjo, Oxherdn, Creepsevry1out, BD2412bot, Fmadd and Anonymous: 77 Vector space Source: https://en.wikipedia.org/wiki/Vector_space?oldid=727289847 Contributors: AxelBoldt, Bryan Derksen, Zundark, The Anome, Taw, Awaterl, Youandme, N8chz, Olivier, Tomo, Patrick, Michael Hardy, Tim Starling, Wshun, Nixdorf, Kku, Gabbe, Wapcaplet, TakuyaMurata, Pcb21, Iulianu, Glenn, Ciphergoth, Dysprosia, Jitse Niesen, Jogloran, Phys, Kwantus, Aenar, Robbot, Romanm, P0lyglut, Tea2min, Giftlite, BenFrantzDale, Lethe, MathKnight, Fropuff, Waltpohl, Andris, Daniel Brockman, Python eggs, Chowbok, Sreyan, Lockeownzj00, MarkSweep, Profvk, Maximaximax, Barnaby dawson, Mh, Klaas van Aarsen, TedPavlic, Rama, Smyth, Notinasnaid, Paul August, Bender235, Rgdboer, Shoujun, Army1987, Cmdrjameson, Stephen Bain, Tsirel, Msh210, Orimosenzon, ChrisUK, Ncik~enwiki, Eric Kvaalen, ABCD, Sligocki, Jheald, Eddie Dealtry, Dirac1933, Woodstone, Kbolino, Oleg Alexandrov, Woohookitty, Mindmatrix, ^demon, Hfarmer, Mpatel, MFH,Graham87, Ilya, Rjwilmsi, Koavf, MarSch, Omnieiunium, Salixalba, Titoxd, FlaBot, VKokielov, Therearenospoons, Nihiltres, Ssafarik, Srleffler, Kri, R160K, Chobot, Gwernol, Algebraist, YurikBot, Wavelength, Spacepotato, Hairy Dude, RussBot, Michael Slone, CambridgeBayWeather, Rick Norwood, Kinser, Guruparan, Trovatore, Vanished user 1029384756, Nick, Bota47, BraneJ, Martinwilke1980, Antiduh, Arthur Rubin, Lonerville, Netrapt, Curpsbot-unicodify, Cjfsyntropy, Paul D. Anderson, GrinBot~enwiki, SmackBot, RDBury, InverseHypercube, KocjoBot~enwiki, Davidsiegel, Chris the speller, SlimJim, SMP, Silly rabbit, Complexica, Nbarth, DHN-bot~enwiki, Colonies Chris, Chlewbot, Vanished User 0001, Cícero, Cybercobra, Daqu, Mattpat, James084, Lambiam, Tbjw, Breno, Terry Bollinger, Frentos, Michael Kinyon, Lim Wei Quan, Rcowlagi, SandyGeorgia, Whackawhackawoo, Inquisitus, Rschwieb, Levineps, JMK, Madmath789, Markan~enwiki, Tawkerbot2, Igni, CRGreathouse, Mct mht, Cydebot, Danman3459, Guitardemon666, Mikewax, Thijs!bot, Headbomb, RobHar, CharlotteWebb, Urdutext, Escarbot, JAnDbot, Thenub314, Englebert, Magioladitis, Jakob.scholbach, Kookas, SwiftBot, WhatamIdoing, David Eppstein, Cpl Syx, Charitwo, Akhil999in, Infovarius, Frenchef, TechnoFaye, CommonsDelinker, Paranomia, Michaelp7, Mitsuruaoyama, Trumpet marietta 45750, Daniele.tampieri, Gombang, Policron, Fylwind, Cartiod, Camrn86, AlnoktaBOT, Hakankösem~enwiki, Belliger~enwiki, TXiKiBoT, Hlevkin, Gwib, Anonymous Dissident, Imasleepviking, Hrrr, Mechakucha, Geometry guy, Terabyte06, Tommyinla, Wikithesource, Staka, AlleborgoBot, Deconstructhis, Newbyguesses, YohanN7, SieBot, Ivan Štambuk, Portalian, ToePeu.bot, Lucasbfrbot, Tiptoety, Paolo.dL, Henry Delforn (old), Thehotelambush, JackSchmidt, Jorgen W, AlanUS, Yoda of Borg, Randomblue, Jludwig, ClueBot, Alksentrs, Nsk92, JP.Martin-Flatin, FractalFusion, Niceguyedc, DifferCake, Auntof6, 0ladne, Bender2k14, PixelBot, Brews ohare, Jotterbot, Hans Adler, SchreiberBike, Jasanas~enwiki, Humanengr, TimothyRias, BodhisattvaBot, SilvonenBot, Jaan Vajakas, Addbot, Gabriele ricci, AndrewHarvey4, Topology Expert, NjardarBot, Looie496, Uncia, ChenzwBot, Ozob, Wikomidia, TeH nOmInAtOr, Jarble, CountryBot, Yobot, Kan8eDie, THEN WHO WAS PHONE?, Edurazee, AnomieBOT, ^musaz, Götz, Citation bot, Xqbot, Txebixev, GeometryGirl, Ferran Mir, Point-set topologist, RibotBOT, Charvest, Quartl, Lisp21, FrescoBot, Nageh, Rckrone, Sławomir Biały, Citation bot 1, Kiefer.Wolfowitz, Jonesey95, MarcelB612, Stpasha, Mathstudent3000, Jujutacular, Dashed, Orenburg1, Double sharp, TobeBot, Javierito92, January, Setitup, TjBot, EmausBot, WikitanvirBot, Brydustin, Fly by Night, Slawekb, Chricho, Ldboer, Quondum, D.Lazard, Milad pourrahmani, RaptureBot, Cloudmichael, Maschen, ClueBot NG, Wcherowi, Chitransh Gaurav, Jiri 1984, Joel B. Lewis, Widr, Helpful Pixie Bot, Ma snx, David815, Alesak23, Probability0001, JOSmithIII, Duxwing, PsiEpsilon, IkamusumeFan, Աննա Շահինյան, IPWAI, JYBot, Dexbot, Catclock, Tch3n93, Fycafterpro, CsDix, Hella.chillz, Jose Brox, François Robere, Loganfalco, Newestcastleman, UY Scuti, K9re11, Monkbot, AntiqueReader, Aditya8795, KurtHeckman, Isambard Kingdom, Shivakrishna .Srinivas. Dasari, NateLloydClark, BowlAndSpoon, Fmadd and Anonymous: 223 Zero divisor Source: https://en.wikipedia.org/wiki/Zero_divisor?oldid=727125556 Contributors: AxelBoldt, Mav, Bryan Derksen, Michael Hardy, TakuyaMurata, Eric119, Andres, Sabbut, SirJective, MathMartin, Tea2min, Giftlite, Dan Gardner, Joseph Dwayne, KarlHenner, MathyGuy23, Vivacissamamente, Mdd, Sabin4232, Graham87, Jshadias, OneWeirdDude, CiaPan, Chobot, Michael Slone, RDBury, Incnis Mrsi, Melchoir, Bluebot, Lambiam, Gleuschk, Rschwieb, CRGreathouse, CmdrObot, Cydebot, Xantharius, Eleuther, Magioladitis, Etale, JoergenB, Jeepday, Trumpet marietta 45750, GaborLajos, Policron, Rei-bot, He7d3r, Carvasf, Feinoha, Astrale, Addbot, Ersik, Jarble, Legobot, Luckas-bot, Yobot, Rubinbot, JackieBot, Citation bot, Erik9bot, Atlantia, DrilBot, Ebony Jackson, Trappist the monk, Sheogorath, RjwilmsiBot, Chricho, Quondum, Savantas83, BG19bot, AvocatoBot, IkamusumeFan, YFdyh-bot, Little green rosetta, Zhongmou Zhang, DrWakewaters, Rayhartung, MatemaatikaLoom, Davyker and Anonymous: 28
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domain Contributors: ? Original artist: ? File:Celticknotwork.png Source: https://upload.wikimedia.org/wikipedia/commons/2/23/Celticknotwork.png License: Public domain Contributors: Transferred from en.wikipedia to Commons by BanyanTree using CommonsHelper. Original artist: The original uploader was CatherineMunro at English Wikipedia File:Cessna_182_model-wingtip-vortex.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/cd/Cessna_182_ model-wingtip-vortex.jpg License: CC-BY-SA-3.0 Contributors: Own work Original artist: BenFrantzDale File:Chance_and_a_Half,_Posing.jpg Source: https://upload.wikimedia.org/wikipedia/commons/c/ca/Chance_and_a_Half%2C_ Posing.jpg License: CC BY 2.0 Contributors: Flickr: Chance and a Half, Posing Original artist: Corinne Cavallo File:Chapitel_IX._of_Die_Theorie_der_algebraischen_Zahlkörper.png Source: https://upload.wikimedia.org/wikipedia/ commons/5/52/Chapitel_IX._of_Die_Theorie_der_algebraischen_Zahlk%C3%B6rper.png License: Public domain Contributors: https://jscholarship.library.jhu.edu/handle/1774.2/34070 Original artist: David Hilbert File:Circle_as_Lie_group2.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/de/Circle_as_Lie_group2.svg License: Public domain Contributors: self-made with en:Inkscape Original artist: Oleg Alexandrov File:Clock_group.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a4/Clock_group.svg License: CC-BY-SA-3.0 Contributors: Transferred from en.wikipedia to Commons. Original artist: The original uploader was Spindled at English Wikipedia File:Commons-logo.svg Source: https://upload.wikimedia.org/wikipedia/en/4/4a/Commons-logo.svg License: CC-BY-SA-3.0 Contributors: ? Original artist: ? File:Commutative_diagram_for_morphism.svg Source: https://upload.wikimedia.org/wikipedia/commons/e/ef/Commutative_ diagram_for_morphism.svg License: Public domain Contributors: Own work, based on en:Image:MorphismComposition-01.png Original artist: User:Cepheus File:Cubane-3D-balls.png Source: https://upload.wikimedia.org/wikipedia/commons/1/18/Cubane-3D-balls.png License: Public domain Contributors: Own work Original artist: Ben Mills File:Cyclic_group.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/5f/Cyclic_group.svg License: CC BY-SA 3.0 Contributors:
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The people from the Tango! project. And according to the meta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (although minimally).” File:Exponentiation_as_monoid_homomorphism_svg.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a3/ Exponentiation_as_monoid_homomorphism_svg.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Jochen Burghardt File:FWF_Samuel_Monnier_détail.jpg Source: https://upload.wikimedia.org/wikipedia/commons/9/99/FWF_Samuel_Monnier_d% C3%A9tail.jpg License: CC BY-SA 3.0 Contributors: Own work (low res file) Original artist: Samuel Monnier File:Farsh1.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/40/Farsh1.jpg License: CC-BY-SA-3.0 Contributors: Transferred from en.wikipedia to Commons. Original artist: Zereshk at English Wikipedia, [1] File:Fifths.png Source: https://upload.wikimedia.org/wikipedia/commons/c/ce/Fifths.png License: CC-BY-SA-3.0 Contributors: ? Original artist: ? File:Folder_Hexagonal_Icon.svg Source: https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg License: Cc-bysa-3.0 Contributors: ? Original artist: ? File:Fundamental_group.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/ba/Fundamental_group.svg License: CC BYSA 3.0 Contributors: en:Image:Fundamental group.png Original artist: en:User:Jakob.scholbach (original); Pbroks13 (talk) (redraw) File:Great_Mosque_of_Kairouan,_west_portico_of_the_courtyard.jpg Source: https://upload.wikimedia.org/wikipedia/commons/ 4/42/Great_Mosque_of_Kairouan%2C_west_portico_of_the_courtyard.jpg License: CC BY-SA 2.0 Contributors: Flickr: marble arch Original artist: James Rose File:Group_D8_180.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/64/Group_D8_180.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_270.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/33/Group_D8_270.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_90.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/f6/Group_D8_90.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_f13.svg Source: https://upload.wikimedia.org/wikipedia/commons/0/0c/Group_D8_f13.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_f24.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a1/Group_D8_f24.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_fh.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b3/Group_D8_fh.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_fv.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/47/Group_D8_fv.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Group_D8_id.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/7e/Group_D8_id.svg License: Public domain Contributors: Own work Original artist: TimothyRias File:Heat_eqn.gif Source: https://upload.wikimedia.org/wikipedia/commons/a/a9/Heat_eqn.gif License: Public domain Contributors: This graphic was created with MATLAB. Original artist: Oleg Alexandrov File:Hexaaquacopper(II)$-$3D-balls.png Source: https://upload.wikimedia.org/wikipedia/commons/0/09/Hexaaquacopper%28II% 29-3D-balls.png License: Public domain Contributors: Own work Original artist: Ben Mills File:Image_Tangent-plane.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/66/Image_Tangent-plane.svg License: Public domain Contributors: Transferred from en.wikipedia to Commons by Ylebru. Original artist: Alexwright at English Wikipedia File:Internet_map_1024.jpg Source: https://upload.wikimedia.org/wikipedia/commons/d/d2/Internet_map_1024.jpg License: CC BY 2.5 Contributors: Originally from the English Wikipedia; description page is/was here. Original artist: The Opte Project File:Irrotationalfield.svg Source: https://upload.wikimedia.org/wikipedia/commons/6/6d/Irrotationalfield.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: AllenMcC. File:Isfahan_Lotfollah_mosque_ceiling_symmetric.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/65/Isfahan_ Lotfollah_mosque_ceiling_symmetric.jpg License: CC BY-SA 3.0 Contributors: Own work Original artist: Phillip Maiwald (Nikopol) File:Kitchen_kaleid.svg Source: https://upload.wikimedia.org/wikipedia/en/c/c3/Kitchen_kaleid.svg License: PD Contributors: ? Original artist: ? File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,1,6,7).svg Source: https://upload.wikimedia.org/wikipedia/ commons/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svgLicense: CCBY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,2,21,23).svg Source: https://upload.wikimedia.org/wikipedia/ commons/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg License: CC BY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,5,14,16).svg Source: https://upload.wikimedia.org/wikipedia/ commons/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg License: CC BY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk) File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg Source: https://upload.wikimedia.org/wikipedia/ commons/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg License: CC BY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)
19.9. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES
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Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist: Tkgd2007 File:Radial_vector_field_dense.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c4/Radial_vector_field_dense.svg License: Public domain Contributors: Self-made, generated with Mathematica 7.0 Original artist: Connor Glosser File:Radial_vector_field_sparse.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b1/Radial_vector_field_sparse.svg License: CC BY-SA 3.0 Contributors: Self-made, generated with Mathematica 7.0 Original artist: Connor Glosser File:Rectangular_hyperbola.svg Source: https://upload.wikimedia.org/wikipedia/commons/2/29/Rectangular_hyperbola.svg License: Public domain Contributors: Own work Original artist: Qef File:Rubik’{}s_cube.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/a6/Rubik%27s_cube.svg License: CC-BY-SA-3.0 Contributors: Based on Image:Rubiks cube.jpg Original artist: This image was created by me, Booyabazooka File:Scalar_multiplication.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/50/Scalar_multiplication.svg License: CC BY 3.0 Contributors: Own work Original artist: Jakob.scholbach File:Sixteenth_stellation_of_icosahedron.png Source: https://upload.wikimedia.org/wikipedia/commons/e/e7/Sixteenth_stellation_ of_icosahedron.png License: CC BY-SA 3.0 Contributors: This image was generated by Vladimir Bulatov’s Polyhedra Stellations Applet: http://bulatov.org/polyhedra/stellation_applet Original artist: Jim2k File:Sphere_symmetry_group_o.svg Source: https://upload.wikimedia.org/wikipedia/commons/3/3a/Sphere_symmetry_group_o.svg License: Public domain Contributors: Made by User:Tomruen Original artist: User:Tomruen File:Studio_del_Corpo_Umano_-_Leonardo_da_Vinci.png Source: https://upload.wikimedia.org/wikipedia/commons/6/69/Studio_ del_Corpo_Umano_-_Leonardo_da_Vinci.png License: Public domain Contributors: Corel Professional Photos CD-ROM Original artist: Leonardo da Vinci File:Symmetric_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,1,14,15,20,21).svg Source: https://upload.wikimedia. org/wikipedia/commons/2/26/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15% 2C20%2C21%29.svg License: CC BY 3.0 Contributors: Own work Original artist: Watchduck (a.k.a. Tilman Piesk)
