Introduction to Abstract Algebra: From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory

583 Pages·2014·4.899 MB·English

by  Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger

Introduction to Abstract Algebra
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Introduction to Abstract Algebra
From Rings, Numbers, Groups, and Fields
to Polynomials and Galois Theory
BENJAMIN FINE
ANTHONY M. GAGLIONE
GERHARD ROSENBERGER
Johns Hopkins University Press
Baltimore
(cid:2)c 2014 Johns Hopkins University Press
All rights reserved. Published 2014
Printed in the United States of America on acid-free paper
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Johns Hopkins University Press
2715 North Charles Street
Baltimore, Maryland 21218-4363
ISBN 13: 978-1-4214-1176-7
ISBN 10: 1-4214-1176-8
Library of Congress Control Number: 2013937859
A catalog record for this book is available from the British Library
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To the memory of Linda and Justin
AMG
To Linda, Carolyn, David, Scott, Jennifer,
Shane, Sawyer, Quinn, Summer, Stratton
BF
To Katariina, Anja, Aila
GR
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Contents
Preface xiii
1 Abstract Algebra and Algebraic Reasoning 1
1.1 Abstract Algebra 1
1.2 Algebraic Structures 2
1.3 The Algebraic Method 3
1.4 The Standard Number Systems 4
1.5 The Integers and Induction 6
1.6 Exercises 10
2 Algebraic Preliminaries 13
2.1 Sets and Set Theory 13
2.1.1 Set Operations 14
2.2 Functions 18
2.3 Equivalence Relations and Factor Sets 21
2.4 Sizes of Sets 25
2.5 Binary Operations 31
2.5.1 The Algebra of Sets 34
2.6 Algebraic Structures and Isomorphisms 36
2.7 Groups 37
2.8 Exercises 41
3 Rings and the Integers 51
3.1 Rings and the Ring of Integers 51
3.2 Some Basic Properties of Rings and Subrings 53
3.3 Examples of Rings 58
3.3.1 The Modular Rings: The Integers Modulo n 59
3.3.2 Noncommutative Rings 63
viii Contents
3.3.3 Rings Without Identities 64
3.3.4 Rings of Subsets: Boolean Rings 65
3.3.5 Direct Sums of Rings 68
3.3.6 Summary of Examples 68
3.4 Ring Homomorphisms and Isomorphisms 70
3.5 Integral Domains and Ordering 72
3.6 Mathematical Induction and the Uniqueness of Z 79
3.7 Exercises 82
4 Number Theory and Unique Factorization 93
4.1 Elementary Number Theory 93
4.2 Divisibility and Primes 97
4.3 Greatest Common Divisors 100
4.4 The Fundamental Theorem of Arithmetic 106
4.5 Congruences and Modular Arithmetic 112
4.6 Unique Factorization Domains 120
4.7 Exercises 121
5 Fields: The Rationals, Reals and Complexes 127
5.1 Fields and Division Rings 127
5.2 Construction and Uniqueness of the Rationals 132
5.2.1 Fields of Fractions 138
5.3 The Real Number System 139
5.3.1 The Completeness of R (Optional) 144
5.3.2 Characterization of R (Optional) 146
5.3.3 The Construction of R (Optional) 146
5.3.4 The p-adic Numbers (Optional) 153
5.4 The Field of Complex Numbers 155
5.4.1 Geometric Interpretation 158
5.4.2 Polar Form and Euler’s Identity 161
5.4.3 DeMoivre’s Theorem for Powers and Roots 165
5.5 Exercises 168
6 Basic Group Theory 181
6.1 Groups, Subgroups and Isomorphisms 181
6.2 Examples of Groups 185
6.2.1 Permutations and the Symmetric Group 187
6.2.2 Examples of Groups: Geometric Transformation Groups 192
6.3 Subgroups and Lagrange’s Theorem 196
Contents ix
6.4 Generators and Cyclic Groups 205
6.5 Exercises 215
7 Factor Groups and the Group Isomorphism Theorems 225
7.1 Normal Subgroups 225
7.2 Factor Groups 233
7.2.1 Examples of Factor Groups 236
7.3 The Group Isomorphism Theorems 242
7.4 Exercises 249
8 Direct Products and Abelian Groups 255
8.1 Direct Products of Groups 255
8.1.1 Direct Products of Two Groups 256
8.1.2 Direct Products of Any Finite Number of Groups 260
8.2 Abelian Groups 266
8.2.1 Finite Abelian Groups 266
8.2.2 Free Abelian Groups 277
8.2.3 The Basis Theorem for Finitely Generated Abelian Groups 282
8.3 Exercises 285
9 Symmetric and Alternating Groups 293
9.1 Symmetric Groups and Cycle Structure 293
9.1.1 The Alternating Groups 299
9.1.2 Conjugation in Sn 302
9.2 The Simplicity of An 303
9.3 Exercises 307
10 Group Actions and Topics in Group Theory 311
10.1 Group Actions 311
10.2 Conjugacy Classes and the Class Equation 313
10.3 The Sylow Theorems 315
10.3.1 Some Applications of the Sylow Theorems 320
10.4 Groups of Small Order 325
10.5 Solvability and Solvable Groups 329
10.5.1 Solvable Groups 330
10.5.2 The Derived Series 336
10.6 Composition Series and the Jordan-Ho¨lder Theorem 338
10.7 Exercises 341