Because many students will not have had much experience with abstract thinking, a number of important concrete examples number theory, integers modulo n, permu-tations are introduced at
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) Algebra, Abstract I Title QA162.N53 2012 512' 02-dc23 2011031416 Printed in the United States of America 10 Contents PREFACE ix ACKNOWLEDGMENTS xvii NOTATION USED IN THE TEXT xix A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 Preliminaries 0.1 0.2 0.3 0.4 Proofs I Sets I Mappings I Equivalences I 17 Integers and Permutations 1.1 1.2 1.3 1.4 1.5 Induction I 24 Divisors and Prime Factorization Integers Modulo n I 42 Permutations I 53 An Application to Cryptography 23 I 32 I 67 Groups 2.1 2.2 2.3 2.4 xxiii Binary Operations I 70 Groups I 76 Subgroups I 86 Cyclic Groups and the Order of an Element 69 I 90 v vi Contents 2.5 2.6 2.8 2.9 2.10 2.11 Polynomials I 203 Factorization of Polynomials Over a Field I 214 Factor Rings of Polynomials Over a Field I 227 Partial Fractions I 236 Symmetric Polynomials I 239 Formal Construction of Polynomials I 248 251 Irreducibles and Unique Factorization Principal Ideal Domains I 264 I 252 Fields 6.1 6.2 6.3 6.4 6.5 6.6 7 202 Factorization in Integral Domains 5.1 5.2 Examples and Basic Properties I 160 Integral Domains and Fields I 171 Ideals and Factor Rings I 180 Homomorphisms I 189 Ordered Integral Domains I 199 Polynomials 4.1 4.2 4.3 4.4 4.5 4.6 159 Rings 3.1 3.2 3.3 3.4 3.5 Homomorphisms and Isomorphisms I 99 Cosets and Lagrange's Theorem I 108 Groups of Motions and Symmetries I 117 Normal Subgroups I 122 Factor Groups I 131 The Isomorphism Theorem I 137 An Application to Binary Linear Codes I 143 274 Vector Spaces I 275 Algebraic Extensions I 283 Splitting Fields I 291 Finite Fields I 298 Geometric Constructions I 304 The Fundamental Theorem of Algebra I 308 An Application to Cyclic and BCH Codes I 310 Modules over Principal Ideal Domains 7.1 7.2 Modules I 324 Modules Over a PID I 335 324 Contents p-Groups and the Sylow Theorems 8.1 8.2 8.3 8.4 8.5 8.6 Products and Factors I 350 Cauchy's Theorem I 357 Group Actions I 364 The Sylow Theorems I 371 Semidirect Products I 379 An Application to Combinatorics I 349 382 Series of Subgroups 9.1 9.2 9.3 The Jordan-Holder Theorem Solvable Groups I 395 Nilpotent Groups 401 388 I 389 10 Galois Theory 10.1 10.2 10.3 10.4 412 Galois Groups and Separability I 413 The Main Theorem of Galois Theory I 422 Insolvability of Polynomials I 434 Cyclotomic Polynomials and Wedderburn's Theorem 11 Finiteness Conditions for Rings and Modules 11.1 Wedderburn's Theorem I 448 11.2 The Wedderburn-Artin Theorem I I 442 447 457 Appendices Appendix Appendix Appendix Appendix vii 71 A Complex Numbers I 471 B Matrix Algebra I 478 C Zorn's Lemma I 486 D Proof of the Recursion Theorem I 490 BIBliOGRAPHY 492 SELECTED ANSWERS 495 INDEX 523 Selected Answers 521 I m E Z and p does not divide m } If z c Y c X, where Y is a subgroup of X, show that there exists ;:;, E Y with n maximal Then show that Y = Z PI, · 11 (a) If x EM show that x- 1r(x) E ker 1r It follows that M = 1r(M)+ ker 1r Continue 13 (a) If K = K1 E& K E& · · • then K ~ K E& K E& •.• ~ K E& K ~ · · · and K1 c K1 E& Kz c K1 E& Kz E& K3 c · · · 15 Use the Hint Let X = {;:;, 17 (a) Show that ker(a)