Abstract algebra / I. N. Herstein.

CONTENIDO
Preface ix
1. Things Familiar and Less Familiar 1
A Few Preliminary Remarks 1
Set Theory 3
Mappings 8
A(S) (The Set of 1-1 Mappings of S onto Itself) 16
The Integers 21
Mathematical Induction 29
Complex Numbers 32
2. Groups 40
Definitions and Examples of Groups 40
Some Simple Remarks 48
Subgroups 51
Lagrange's Theorem 56
Homomorphisms and Normal Subgroups 66
Factor Groups 77
The Homomorphism Theorems 84
Cauchy's Theorem 88
Direct Products 92
Finite Abelian Groups (Optional) 96
Conjugacy and Sylow's Theorem (Optional) 101
3. The Symmetric Group 108
Preliminaries 108
Cycle Decomposition 111
Odd and Even Permutations 119
4. Ring Theory 125
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
5. Fields 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
The Simplicity of An 215
Finite Fields I 221
Finite Fields II: Existence 224
Finite Fields III: Uniqueness 227
Cyclotomic Polynomials 229
Liouville's Criterion 236
The Irrationality of Pi 239
Index 243