Elements of Modern Algebra

524 Pages·2010·2.3 MB·English

by  Linda Gilbert

Elements of
Modern Algebra
This page intentionally left blank
S E V E N T H E D I T I O N
Elements of
Modern Algebra
Linda Gilbert
University of South Carolina Upstate
Jimmie Gilbert
Late of University of South Carolina Upstate
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Elements of Modern Algebra, © 2009, 2005Brooks/Cole, Cengage Learning
Seventh Edition
ALL RIGHTS RESERVED. No part of this work covered by the copyright
Linda Gilbert, Jimmie Gilbert
herein may be reproduced, transmitted, stored, or used in any form or by
Editor: Molly Taylor any means graphic, electronic, or mechanical, including but not limited to
Development Editor: Stacy Green photocopying, recording, scanning, digitizing, taping, Web distribution, in-
formation networks, or information storage and retrieval systems, except
Assistant Editor: Dan Seibert as permitted under Section 107or 108of the 1976United States Copyright
Editorial Assistant: Cynthia Ashton Act, without the prior written permission of the publisher.
Marketing Specialist: Ashley Pickering
Marketing Assistant: Angela Kim For product information and technology assistance, contact us at
Cengage Learning Customer & Sales Support, 1-800-354-9706.
Marketing Communications Manager:
Mary Anne Payumbo For permission to use material from this text or product,
submit all requests online at www.cengage.com/permissions.
Project Manager, Editorial Production: Further permissions questions can be e-mailed to
Cheryll Linthicum [email protected].
Creative Director: Rob Hugel
Art Director: Vernon Boes Library of Congress Control Number: 2008928823
Print Buyer: Paula Vang ISBN-13: 978-0495-56136-1
Permissions Editor: Tim Sisler
ISBN-10: 0-495-56136-3
Production Service: ICC Macmillan Inc.
Text Designer: Roy Neuhaus
Brooks/Cole
Photo Researcher: Martha Hall 10Davis Drive
Copy Editor: AmyLyn Reynolds Belmont, CA 94002-3098
Illustrator: ICC Macmillan Inc. USA
Cover Designer: Matt Gilbert
Cover Image: Marsha Cohen Cengage Learning is a leading provider of customized learning solutions
Compositor: ICC Macmillan Inc. with office locations around the globe, including Singapore, the United
Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at
international.cengage.com/region.
Cengage Learning products are represented in Canada by Nelson
Education, Ltd.
For your course and learning solutions, visit academic.cengage.com.
Purchase any of our products at your local college store or at our preferred
online store www.ichapters.com.
Printed in the United States of America
1 2 3 4 5 6 7 12 11 10 09 08
To: Jimmie
~~Linda
This page intentionally left blank
Contents
Preface xi
1
Fundamentals 1
1.1 Sets 1
1.2 Mappings 12
1.3 Properties of Composite Mappings (Optional) 25
1.4 Binary Operations 30
1.5 Permutations and Inverses 37
1.6 Matrices 42
1.7 Relations 55
Key Words and Phrases 62
A Pioneer in Mathematics: Arthur Cayley 62
2
The Integers 65
2.1 Postulates for the Integers (Optional) 65
2.2 Mathematical Induction 71
2.3 Divisibility 81
2.4 Prime Factors and Greatest Common Divisor 86
2.5 Congruence of Integers 95
2.6 Congruence Classes 107
2.7 Introduction to Coding Theory (Optional) 114
2.8 Introduction to Cryptography (Optional) 123
Key Words and Phrases 134
A Pioneer in Mathematics: Blaise Pascal 135
3
Groups 137
3.1 Definition of a Group 137
3.2 Properties of Group Elements 145
vii
viii Contents
3.3 Subgroups 152
3.4 Cyclic Groups 163
3.5 Isomorphisms 174
3.6 Homomorphisms 183
Key Words and Phrases 188
A Pioneer in Mathematics: Niels Henrik Abel 189
4
More on Groups 191
4.1 Finite Permutation Groups 191
4.2 Cayley’s Theorem 205
4.3 Permutation Groups in Science and Art (Optional) 208
4.4 Cosets of a Subgroup 215
4.5 Normal Subgroups 223
4.6 Quotient Groups 230
4.7 Direct Sums (Optional) 239
4.8 Some Results on Finite Abelian Groups (Optional) 246
Key Words and Phrases 255
A Pioneer in Mathematics: Augustin Louis Cauchy 256
5
Rings, Integral Domains, and Fields 257
5.1 Definition of a Ring 257
5.2 Integral Domains and Fields 270
5.3 The Field of Quotients of an Integral Domain 276
5.4 Ordered Integral Domains 284
Key Words and Phrases 291
A Pioneer in Mathematics: Richard Dedekind 292
6
More on Rings 293
6.1 Ideals and Quotient Rings 293
6.2 Ring Homomorphisms 303
6.3 The Characteristic of a Ring 313
6.4 Maximal Ideals (Optional) 319
Key Words and Phrases 324
A Pioneer in Mathematics: Amalie Emmy Noether 324
Contents ix
7
Real and Complex Numbers 325
7.1 The Field of Real Numbers 325
7.2 Complex Numbers and Quaternions 333
7.3 De Moivre’s Theorem and Roots of Complex Numbers 343
Key Words and Phrases 352
A Pioneer in Mathematics: William Rowan Hamilton 353
8
Polynomials 355
8.1 Polynomials over a Ring 355
8.2 Divisibility and Greatest Common Divisor 367
8.3 Factorization in F[x] 375
8.4 Zeros of a Polynomial 384
8.5 Solution of Cubic and Quartic Equations by Formulas (Optional) 397
8.6 Algebraic Extensions of a Field 409
Key Words and Phrases 421
A Pioneer in Mathematics: Carl Friedrich Gauss 422
The Basics of Logic 423
APPENDIX:
Answers to True/False and Selected
Computational Exercises 435
Bibliography 499
Index 503