Contemporary Abstract Algebra

654 Pages·2020·28.378 MB·English

by  Joseph A. Gallian

Contemporary
Abstract Algebra
Textbooks in Mathematics
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Contemporary
Abstract Algebra
TENTH EDITION
Joseph A. Gallian
University of Minnesota Duluth
Cover image by John Stembridge
2-dimensional representation of E8 Lie group
First edition published 1986 D. C. Heath and Company
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and by CRC Press
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Library of Congress Cataloging‑in‑Publication Data
ISBN: 978-0-367-65178-7 (hbk)
ISBN: 978-1-003-14233-1 (ebk)
Typeset in Computer Modern font
by KnowledgeWorks Global Ltd.
To Char, Krissy, and Joey

Contents
Notations xiii
Preface xvii
0 Preliminaries 1
Properties of Integers 1 | Modular Arithmetic 5 | Complex
Numbers 10 | Mathematical Induction 12 | Equivalence
Relations 15|Functions (Mappings) 18
Exercises 21
1 Introduction to Groups 27
Symmetries of a Square 27|The Dihedral Groups 30
Biography of Niels Abel 38
2 Groups 39
Definition and Examples of Groups 39|Elementary Properties
of Groups 47|Historical Note 50
Exercises 52
3 Finite Groups; Subgroups 59
Terminology and Notation 59|Subgroup Tests 61|Examples
of Subgroups 64
Exercises 69
4 Cyclic Groups 77
Properties of Cyclic Groups 77|Classification of Subgroups of
Cyclic Groups 83
Biography of James Joseph Sylvester 96
5 Permutation Groups 97
Definitions and Notation 97|Cycle Notation 100|Properties
of Permutations 103|A Check-Digit Scheme Based on D 115
5
vii
viii Contents
Biography of Augustin Cauchy 125
Biography of Alan Turing 126
6 Isomorphisms 127
Motivation 127 | Definition and Examples 127 | Properties of
Isomorphisms131|Automorphisms134|Cayley’sTheorem138
Exercises 141
Biography of Arthur Cayley 147
7 Cosets and Lagrange’s Theorem 149
Properties of Cosets 149|Lagrange’s Theorem and
Consequences 153|An Application of Cosets to Permutation
Groups 158 | The Rotation Group of a Cube and a Soccer
Ball 160|An Application of Cosets to the Rubik’s Cube 163
Exercises 163
Biography of Joseph Lagrange 170
8 External Direct Products 171
Definition and Examples 171 | Properties of External Direct
Products 173|The Group of Units Modulo n as an External
Direct Product 176|Applications 178
Exercises 184
Biography of Leonard Adleman 191
9 Normal Subgroups and Factor Groups 193
Normal Subgroups 193 | Factor Groups 196 | Applications of
Factor Groups 200|Internal Direct Products 203
Exercises 209
Biography of Évariste Galois 204
10 Group Homomorphisms 219
DefinitionandExamples219|PropertiesofHomomorphisms221
|The First Isomorphism Theorem 225
Exercises 232
Biography of Camille Jordan 239
11 Fundamental Theorem of Finite Abelian Groups 241
The Fundamental Theorem 241|The Isomorphism Classes of
Abelian Groups 242|Proof of the Fundamental Theorem 246
Exercises 249
Contents ix
12 Introduction to Rings 255
Motivation and Definition 255 | Examples of Rings 256 |
Properties of Rings 257|Subrings 259
Exercises 261
Biography of I. N. Herstein 266
13 Integral Domains 267
Definition and Examples 267|Fields 268|Characteristic of a
Ring 271
Exercises 273
14 Ideals and Factor Rings 279
Ideals 279 | Factor Rings 280 | Prime Ideals and Maximal
Ideals 284
Exercises 286
Biography of Richard Dedekind 293
Biography of Emmy Noether 294
15 Ring Homomorphisms 295
Definition and Examples 295|Properties of Ring
Homomorphisms 298|The Field of Quotients 301
Exercises 303
16 Polynomial Rings 311
Notation and Terminology 311|The Division Algorithm and
Consequences 314
Exercises 319
17 Factorization of Polynomials 325
Reducibility Tests 325|Irreducibility Tests 328|Unique Fac-
torization in Z[x] 334|Weird Dice: An Application of Unique
Factorization 335
Exercises 338
Biography of Serge Lang 342
18 Divisibility in Integral Domains 343
Irreducibles, Primes 343 | Historical Discussion of Fermat’s
Last Theorem 346 | Unique Factorization Domains 350 |
Euclidean Domains 353
Exercises 356
Biography of Sophie Germain 361