Abstract Algebra

CONTENTS
Preface ix
1 Things Familiar and Less Familiar 1
1 A Few Preliminary Remarks 1
2 Set Theory 3
3 Mappings 8
4 A(S) (The Set of 1-1 Mappings of S onto Itself) 16
5 The Integers 21
6 Mathematical Induction 29
7 Complex Numbers 32
2 Groups 40
1 Definitions and Examples of Groups 40
2 Some Simple Remarks 48
3 Subgroups 51
4 Lagrange's Theorem 56
5 Homomorphisms and Normal Subgroups 66
6 Factor Groups 77
7 The Homomorphism Theorems 84
8 Cauchy's Theorem 88
9 Direct Products 92
10 Finite Abelian Groups (Optional) 96
11 Conjugacy and Sylow's Theorem (Optional) 101
3 The Symmetric Group 108
1 Preliminaries 108
2 Cycle Decomposition 111
3 Odd and Even Permutations 119
4 Ring Theory 125
1
2
3
4
5
6
7
5 Fields
1
2
3
4
5
6
Definitions and Examples 125
Some Simple Results 137
Ideals, Homomorphisms, and Quotient Rings 139
Maximal Ideals 148
Polynomial Rings 151
Polynomials over the Rationals 166
Field of Quotients of an Integral Domain 172
176
Examples of Fields 176
A Brief Excursion into Vector Spaces 180
Field Extensions 191
Finite Extensions 198
Constructibility 201
Roots of Polynomials 207
6 Special Topics (Optional) 215
1 The Simplicity of An 215
2 Finite Fields I 221
3 Finite Fields II: Existence 224
4 Finite Fields III: Uniqueness 227
5 Cyclotomic Polynomials 229
6 Liouville's Criterion 236
7 The Irrationality of 'IT 239
Index 243

Another The Core Of CS
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