Introduction to Abstract Algebra

561 Pages·2012·23.82 MB·English

by  W. Keith Nicholson

1\flWILEY

Introduction to
Abstract Algebra
. I
Introduction to
Abstract Algebra
Fourth Edition
W. Keith Nicholson
University of Calgary
Calgary, Alberta, Canada
@)WILEY
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright 2012 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Nicholson, W. Keith.
Introduction to abstract algebra / W. Keith Nicholson. - 4th ed.
p. em.
Includes bibliographical references and index.
ISBN 978-1-118-13535-8 (cloth)
1. Algebra, Abstract. I. Title.
QA162.N53 2012
512' .02-dc23 2011031416
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
Contents
PREFACE ix
ACKNOWLEDGMENTS xvii
NOTATION USED IN THE TEXT xix
A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii
0 Preliminaries 1
0.1 Proofs I 1
0.2 Sets I 5
0.3 Mappings I 9
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0.4 Equivalences 17
1 Integers and Permutations 23
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1.1 Induction 24
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1.2 Divisors and Prime Factorization 32
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1.3 Integers Modulo n 42
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1.4 Permutations 53
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1.5 An Application to Cryptography 67
2 Groups 69
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2.1 Binary Operations 70
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2.2 Groups 76
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2.3 Subgroups 86
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2.4 Cyclic Groups and the Order of an Element 90
v
vi Contents
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2.5 Homomorphisms and Isomorphisms 99
2.6 Cosets and Lagrange's Theorem I 108
2. 7 Groups of Motions and Symmetries I 117
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2.8 Normal Subgroups 122
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2.9 Factor Groups 131
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2.10 The Isomorphism Theorem 137
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2.11 An Application to Binary Linear Codes 143
3 Rings 159
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3.1 Examples and Basic Properties 160
3.2 Integral Domains and Fields I 171
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3.3 Ideals and Factor Rings 180
3.4 Homomorphisms I 189
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3.5 Ordered Integral Domains 199
4 Polynomials 202
4.1 Polynomials I 203
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4.2 Factorization of Polynomials Over a Field 214
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4.3 Factor Rings of Polynomials Over a Field 227
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4.4 Partial Fractions 236
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4.5 Symmetric Polynomials 239
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4.6 Formal Construction of Polynomials 248
5 Factorization in Integral Domains 251
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5.1 Irreducibles and Unique Factorization 252
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5.2 Principal Ideal Domains 264
6 Fields 274
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6.1 Vector Spaces 275
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6.2 Algebraic Extensions 283
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6.3 Splitting Fields 291
6.4 Finite Fields I 298
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6.5 Geometric Constructions 304
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6.6 The Fundamental Theorem of Algebra 308
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6. 7 An Application to Cyclic and BCH Codes 310
7 Modules over Principal Ideal Domains 324
7.1 Modules I 324
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7.2 Modules Over a PID 335
Contents vii
8 p-Groups and the Sylow Theorems 349
8.1 Products and Factors I 350
8.2 Cauchy's Theorem I 357
8.3 Group Actions I 364
8.4 The Sylow Theorems I 371
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8.5 Semidirect Products 379
8.6 An Application to Combinatorics I 382
9 Series of Subgroups 388
9.1 The Jordan-Holder Theorem I 389
9.2 Solvable Groups I 395
9.3 Nilpotent Groups 1 401
10 Galois Theory 412
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10.1 Galois Groups and Separability 413
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10.2 The Main Theorem of Galois Theory 422
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10.3 Insolvability of Polynomials 434
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10.4 Cyclotomic Polynomials and Wedderburn's Theorem 442
11 Finiteness Conditions for Rings and Modules 447
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11.1 Wedderburn's Theorem 448
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11.2 The Wedderburn-Artin Theorem 457
Appendices 4 71
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Appendix A Complex Numbers 471
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Appendix B Matrix Algebra 478
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Appendix C Zorn's Lemma 486
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Appendix D Proof of the Recursion Theorem 490
BIBliOGRAPHY 492
SELECTED ANSWERS 495
INDEX 523