Abstract Algebra

Cover......Page 1
Title page......Page 3
Date-line......Page 4
Dedication......Page 5
CONTENTS......Page 7
Preface to the third edition......Page 9
Preface to the first edition......Page 11
Symbol list......Page 13
1 A Few Preliminary Remarks......Page 17
2 Set Theory......Page 19
3 Mappings......Page 24
4 $A(S)$ (The Set of 1-1 Mappings of $S$ onto Itself)......Page 32
5 The Integers......Page 37
6 Mathematical Induction......Page 45
7 Complex Numbers......Page 48
1 Definitions and Examples of Groups......Page 56
2 Some Simple Remarks......Page 64
3 Subgroups......Page 67
4 Lagrange\'s Theorem......Page 72
5 Homomorphisms and Normal Subgroups......Page 82
6 Factor Groups......Page 93
7 The Homomorphism Theorems......Page 100
8 Cauchy\'s Theorem......Page 104
9 Direct Products......Page 108
10 Finite Abelian Groups (Optional)......Page 112
11 Conjugacy and Sylow\'s Theorem (Optional)......Page 117
1 Preliminaries......Page 124
2 Cycle Decomposition......Page 127
3 Odd and Even Permutations......Page 135
1 Definitions and Examples......Page 141
2 Some Simple Results......Page 153
3 Ideals, Homomorphisms, and Quotient Rings......Page 155
4 Maximal Ideals......Page 164
5 Polynomial Rings......Page 167
6 Polynomials over the Rationals......Page 182
7 Field of Quotients of an Integral Domain......Page 188
1 Examples of Fields......Page 192
2 A Brief Excursion into Vector Spaces......Page 196
3 Field Extensions......Page 207
4 Finite Extensions......Page 214
5 Constructibility......Page 217
6 Roots of Polynomials......Page 223
1 The Simplicity of An......Page 231
2 Finite Fields I......Page 237
3 Finite Fields II: Existence......Page 240
4 Finite Fields III: Uniqueness......Page 243
5 Cyclotomic Polynomials......Page 245
6 Liouville\'s Criterion......Page 252
7 The Irrationality of $\\pi$......Page 255
Index......Page 259