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. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability /Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department with the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. 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 I 0.4 Equivalences 17 1 Integers and Permutations 23 I 1.1 Induction 24 I 1.2 Divisors and Prime Factorization 32 I 1.3 Integers Modulo n 42 I 1.4 Permutations 53 I 1.5 An Application to Cryptography 67 2 Groups 69 I 2.1 Binary Operations 70 I 2.2 Groups 76 I 2.3 Subgroups 86 I 2.4 Cyclic Groups and the Order of an Element 90 v vi Contents I 2.5 Homomorphisms and Isomorphisms 99 2.6 Cosets and Lagrange's Theorem I 108 2. 7 Groups of Motions and Symmetries I 117 I 2.8 Normal Subgroups 122 I 2.9 Factor Groups 131 I 2.10 The Isomorphism Theorem 137 I 2.11 An Application to Binary Linear Codes 143 3 Rings 159 I 3.1 Examples and Basic Properties 160 3.2 Integral Domains and Fields I 171 I 3.3 Ideals and Factor Rings 180 3.4 Homomorphisms I 189 I 3.5 Ordered Integral Domains 199 4 Polynomials 202 4.1 Polynomials I 203 I 4.2 Factorization of Polynomials Over a Field 214 I 4.3 Factor Rings of Polynomials Over a Field 227 I 4.4 Partial Fractions 236 I 4.5 Symmetric Polynomials 239 I 4.6 Formal Construction of Polynomials 248 5 Factorization in Integral Domains 251 I 5.1 Irreducibles and Unique Factorization 252 I 5.2 Principal Ideal Domains 264 6 Fields 274 I 6.1 Vector Spaces 275 I 6.2 Algebraic Extensions 283 I 6.3 Splitting Fields 291 6.4 Finite Fields I 298 I 6.5 Geometric Constructions 304 I 6.6 The Fundamental Theorem of Algebra 308 I 6. 7 An Application to Cyclic and BCH Codes 310 7 Modules over Principal Ideal Domains 324 7.1 Modules I 324 I 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 I 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 I 10.1 Galois Groups and Separability 413 I 10.2 The Main Theorem of Galois Theory 422 I 10.3 Insolvability of Polynomials 434 I 10.4 Cyclotomic Polynomials and Wedderburn's Theorem 442 11 Finiteness Conditions for Rings and Modules 447 I 11.1 Wedderburn's Theorem 448 I 11.2 The Wedderburn-Artin Theorem 457 Appendices 4 71 I Appendix A Complex Numbers 471 I Appendix B Matrix Algebra 478 I Appendix C Zorn's Lemma 486 I Appendix D Proof of the Recursion Theorem 490 BIBliOGRAPHY 492 SELECTED ANSWERS 495 INDEX 523