ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONS
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ABSTRACT ALGEBRA MANUAL: PROBLEMS AND SOLUTIONS
AYMAN BADAWI
Nova Science Publishers, Inc. Huntington, NY
Senior Editors: Susan Boriotti and Donna Dennis Editorial Coordinator: Tatiana Shohov Office Manager: Annette Hellinger Graphics: Wanda Serrano Book Production: Matthew Kozlowski, Jonathan Rose and Jennifer Vogt Circulation: Cathy DeGregory, Ave Maria Gonzalez, Ron Hedges and Andre Tillman
Library of Congress Cataloging-in-Publication Data Badawi, Ayman. Abstract algebra manual: problems and solutions / Ayman Badawi p. cm. Includes bibliographical references and index. ISBN:
(eBook)
1. Algebra, Abstract. I. Title. QA162 .B33 2001 512’.02— dc21 2001031272
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INDEX
A Abelian, 5 automorphisms, 2 B Binomial Expansion Theorem, 68 C cancelation, left-, 32 cancelation, right-, 32 centralizer, 1 closed, 3 closure, 11 common divisor, greatest, 1 common divisor, least, 1 common entries, 4 complex numbers, all nonzero, 29 contradiction, 11 cosets, 1 cosets, left, 21 cycle, even, 19 cycles, disjoint, 4, 18 cyclic group, infinite, 35 cyclic, 3
D distinct powers, 3 distinct prime numbers, 3 divisor, 3 divisor, positive, 4 domain, 62 domain, integral, 65 domain, principal ideal, 69 E element, 2 element, nilpotent, 62 element, non-identity, 8, 49 elements, distinct, 3, 13 elements, nilpotent, 55 F field, 56 field, extension, 58, 93 field, finite extension, 58 field, finite, 59 field, infinite, 85 field, some extension, 96 field, splitting, 92 finite group, 4
108 finite index, 29 finite set, 2 finite, order, 3 G generators, 16 group homomorphism, 5, 34 group homomorphism, nontrivial, 37 group isomorphism, 36 group, Abelian, 13 group, cyclic, 3 group, finite Abelian, 36 group, infinite cyclic, 16 group, infinite, 17 group, non-Abelian, 24, 47 group, non-cyclic Abelian, 17 group, non-cyclic, 25 group, unit, 2 group-axioms, 11 H homomorphism, ring, 34 I ideal , maximal, 67 ideal domain, principle, 56 ideal, nonzero prime, 88 ideal, prime, 65 ideal, principal, 67 ideal, proper, 66 ideals, all prime (maximal), 88 ideals, prime (maximal), 81 idempotent, 62 identity matrix, 63 infinite order, 3, 15, 85 integer, even, 70 integer, odd, 70 integer, positive, 4 integers, 27 integral domain, 55, 73
A. Badawi integral domain, finite 73 integer, positive, 2, 31 integers, 2 isomorphic, 2 isomorphisms, 2 L least common multiple, 4 linear factors, 90 M matrix, square, 2 matrix-multiplication, 2 maximal ideal, 56, 57 maximal order, 53 module, multiplication, 2 module, intergers, 2 multiplicative identity, 77 multiplicity, 86 N nilpotent, 63 non-Albelian, 45 nonzero-ring homomorphism, 83 normalizer, 1 number, prime, 2, 23 numbers, all rational, 55 numbers, all real, 55 numbers, complex, 2 numbers, rational, 2 O ONTO, 80, P permutation of a finite set, 4 permutations, 2 polynomial, 55 polynomial, irreducible, 59, 90
