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16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page i Contemporary Abstract Algebra SEVENTH EDITION Joseph A. Gallian University of Minnesota Duluth Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page ii Contemporary Abstract Algebra, © 2010,2006Brooks/Cole, Cengage Learning Seventh Edition ALL RIGHTS RESERVED. No part of this work covered by the Joseph A. Gallian copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or VP/Editor-in-Chief: Michelle Julet mechanical, including but not limited to photocopying, Publisher: Richard Stratton recording, scanning, digitizing, taping, Web distribution, Senior Sponsoring Editor: Molly Taylor information networks, or information storage and retrieval systems, except as permitted under Section 107or 108of Associate Editor: Daniel Seibert the1976United States Copyright Act, without the prior Editorial Assistant: Shaylin Walsh written permission of the publisher. Managing Media Editor: Sam Subity Senior Content Manager: Maren Kunert For product information and Executive Marketing Manager: Joe Rogove technology assistance, contact us at Cengage Learning Customer & Sales Support,1-800-354-9706 Marketing Specialist: Ashley Pickering Marketing Communications Manager: For permission to use material from this text Mary Anne Payumo or product, submit all requests online at www.cengage.com/permissions.Further permissions Senior Content Project Manager, Editorial questions can be e-mailed to Production: Tamela Ambush [email protected]. Senior Manufacturing Coordinator: Diane Gibbons Senior Rights Acquisition Account Manager: Library of Congress Control Number: 2008940386 Katie Huha Student Edition: Production Service: Matrix Productions Inc. ISBN-13:978-0-547-16509-7 ISBN-10:0-547-16509-9 Text Designer: Ellen Pettengell Design Photo Researcher: Lisa Jelly Smith Brooks/Cole Cover Designer: Elise Vandergriff 10 Davis Drive Belmont, CA 94002-3098 Cover Image: © Anne M. Burns USA Compositor: Pre-PressPMG Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/international. Cengage Learning products are represented in Canada by Nelson Education, Ltd. To learn more about Brooks/Cole, visit www.cengage.com/brookscole. 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 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page iii Contents Preface xi Integers and Equivalence Relations 1 PART 1 0 Preliminaries 3 Properties of Integers 3 | Modular Arithmetic 7 | Mathematical Induction 12 | Equivalence Relations 15 | Functions(Mappings) 18 Exercises 21 Computer Exercises 25 Groups 27 PART 2 1 Introduction to Groups 29 Symmetries of a Square 29 | The Dihedral Groups 32 Exercises 35 Biography of Niels Abel 39 2 Groups 40 Definition and Examples of Groups 40 | Elementary Properties of Groups 48 | Historical Note 51 Exercises 52 Computer Exercises 55 3 Finite Groups; Subgroups 57 Terminology and Notation 57 | Subgroup Tests 58 | Examples of Subgroups 61 Exercises 64 Computer Exercises 70 iii 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page iv iv Contents 4 Cyclic Groups 72 Properties of Cyclic Groups 72 | Classification of Subgroups of Cyclic Groups 77 Exercises 81 Computer Exercises 86 Biography of J. J. Sylvester 89 Supplementary Exercises for Chapters 1–4 91 5 Permutation Groups 95 Definition and Notation 95 | Cycle Notation 98 | Properties of Permutations 100 | A Check Digit Scheme Based on D 110 5 Exercises 113 Computer Exercises 118 Biography of Augustin Cauchy 121 6 Isomorphisms 122 Motivation 122 | Definition and Examples 122 | Cayley’s Theorem 126 | Properties of Isomorphisms 128 | Automorphisms 129 Exercises 133 Computer Exercise 136 Biography of Arthur Cayley 137 7 Cosets and Lagrange’s Theorem 138 Properties of Cosets 138 | Lagrange’s Theorem and Consequences 141 | An Application of Cosets to Permutation Groups 145 | The Rotation Group of a Cube and a Soccer Ball 146 Exercises 149 Computer Exercise 153 Biography of Joseph Lagrange 154 8 External Direct Products 155 Definition and Examples 155 | Properties of External Direct Products 156 | The Group of Units Modulo nas an External Direct Product 159 | Applications 161 Exercises 167 Computer Exercises 170 Biography of Leonard Adleman 173 Supplementary Exercises for Chapters 5–8 174 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page v Contents v 9 Normal Subgroups and Factor Groups 178 Normal Subgroups 178 | Factor Groups 180 | Applications of Factor Groups 185 | Internal Direct Products 188 Exercises 193 Biography of Évariste Galois 199 10 Group Homomorphisms 200 Definition and Examples 200 | Properties of Homomorphisms 202 | The First Isomorphism Theorem 206 Exercises 211 Computer Exercise 216 Biography of Camille Jordan 217 11 Fundamental Theorem of Finite Abelian Groups 218 The Fundamental Theorem 218 | The Isomorphism Classes of Abelian Groups 218 | Proof of the Fundamental Theorem 223 Exercises 226 Computer Exercises 228 Supplementary Exercises for Chapters 9–11 230 Rings 235 PART 3 12 Introduction to Rings 237 Motivation and Definition 237 | Examples of Rings 238 | Properties of Rings 239 | Subrings 240 Exercises 242 Computer Exercises 245 Biography of I. N. Herstein 248 13 Integral Domains 249 Definition and Examples 249 | Fields 250 | Characteristic of a Ring 252 Exercises 255 Computer Exercises 259 Biography of Nathan Jacobson 261 14 Ideals and Factor Rings 262 Ideals 262 | Factor Rings 263 | Prime Ideals and Maximal Ideals 267 Exercises 269 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page vi vi Contents Computer Exercises 273 Biography of Richard Dedekind 274 Biography of Emmy Noether 275 Supplementary Exercises for Chapters 12–14 276 15 Ring Homomorphisms 280 Definition and Examples 280 | Properties of Ring Homomorphisms 283 | The Field of Quotients 285 Exercises 287 16 Polynomial Rings 293 Notation and Terminology 293 | The Division Algorithm and Consequences 296 Exercises 300 Biography of Saunders Mac Lane 304 17 Factorization of Polynomials 305 Reducibility Tests 305 | Irreducibility Tests 308 | Unique Factorization in Z[x] 313 | Weird Dice:An Application of Unique Factorization 314 Exercises 316 Computer Exercises 319 Biography of Serge Lang 321 18 Divisibility in Integral Domains 322 Irreducibles,Primes 322 | Historical Discussion of Fermat’s Last Theorem 325 | Unique Factorization Domains 328 | Euclidean Domains 331 Exercises 335 Computer Exercise 337 Biography of Sophie Germain 339 Biography of Andrew Wiles 340 Supplementary Exercises for Chapters 15–18 341 Fields 343 PART 4 19 Vector Spaces 345 Definition and Examples 345 | Subspaces 346 | Linear Independence 347 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page vii Contents vii Exercises 349 Biography of Emil Artin 352 Biography of Olga Taussky-Todd 353 20 Extension Fields 354 The Fundamental Theorem of Field Theory 354 | Splitting Fields 356 | Zeros of an Irreducible Polynomial 362 Exercises 366 Biography of Leopold Kronecker 369 21 Algebraic Extensions 370 Characterization of Extensions 370 | Finite Extensions 372 | Properties of Algebraic Extensions 376 | Exercises 378 Biography of Irving Kaplansky 381 22 Finite Fields 382 Classification of Finite Fields 382 | Structure of Finite Fields 383 | Subfields of a Finite Field 387 Exercises 389 Computer Exercises 391 Biography of L. E. Dickson 392 23 Geometric Constructions 393 Historical Discussion of Geometric Constructions 393 | Constructible Numbers 394 | Angle-Trisectors and Circle-Squarers 396 Exercises 396 Supplementary Exercises for Chapters 19–23 399 Special Topics 401 PART 5 24 Sylow Theorems 403 Conjugacy Classes 403 | The Class Equation 404 | The Probability That Two Elements Commute 405 | The Sylow Theorems 406 | Applications of Sylow Theorems 411 Exercises 414 Computer Exercise 418 Biography of Ludwig Sylow 419 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page viii viii Contents 25 Finite Simple Groups 420 Historical Background 420 | Nonsimplicity Tests 425 | The Simplicity of A 429 | The Fields Medal 430 | 5 The Cole Prize 430 | Exercises 431 Computer Exercises 432 Biography of Michael Aschbacher 434 Biography of Daniel Gorenstein 435 Biography of John Thompson 436 26 Generators and Relations 437 Motivation 437 | Definitions and Notation 438 | Free Group 439 | Generators and Relations 440 | Classification of Groups of Order Up to 15 444 | Characterization of Dihedral Groups 446 | Realizing the Dihedral Groups with Mirrors 447 Exercises 449 Biography of Marshall Hall,Jr. 452 27 Symmetry Groups 453 Isometries 453 | Classification of Finite Plane Symmetry Groups 455 | Classification of Finite Groups of Rotations in R3 456 Exercises 458 28 Frieze Groups and Crystallographic Groups 461 The Frieze Groups 461 | The Crystallographic Groups 467 | Identification of Plane Periodic Patterns 473 Exercises 479 Biography of M. C. Escher 484 Biography of George Pólya 485 Biography of John H. Conway 486 29 Symmetry and Counting 487 Motivation 487 | Burnside’s Theorem 488 | Applications 490 | Group Action 493 Exercises 494 Biography of William Burnside 497 30 Cayley Digraphs of Groups 498 Motivation 498 | The Cayley Digraph of a Group 498 | Hamiltonian Circuits and Paths 502 | Some Applications 508 16509_fm_pi-xiv pp2.qxd 11/20/08 4:35 AM Page ix Contents ix Exercises 511 Biography of William Rowan Hamilton 516 Biography of Paul Erdös 517 31 Introduction to Algebraic Coding Theory 518 Motivation 518 | Linear Codes 523 | Parity-Check Matrix Decoding 528 | Coset Decoding 531 | Historical Note: The Ubiquitous Reed-Solomon Codes 535 Exercises 537 Biography of Richard W. Hamming 542 Biography of Jessie MacWilliams 543 Biography of Vera Pless 544 32 An Introduction to Galois Theory 545 Fundamental Theorem of Galois Theory 545 | Solvability of Polynomials by Radicals 552 | Insolvability of a Quintic 556 Exercises 557 Biography of Philip Hall 560 33 Cyclotomic Extensions 561 Motivation 561 | Cyclotomic Polynomials 562 | The Constructible Regular n-gons 566 Exercises 568 Computer Exercise 569 Biography of Carl Friedrich Gauss 570 Biography of Manjul Bhargava 571 Supplementary Exercises for Chapters 24–33 572 Selected Answers A1 Text Credits A40 Photo Credits A42 Index of Mathematicians A43 Index of Terms A45

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Contemporary Abstract Algebra University of Minnesota Duluth Publisher: Richard Stratton Senior Sponsoring Editor: Molly Taylor
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