CONTENTS
CHAPTER 1 Things familiar and Less Familiar = 1
1 A Few Preliminary Remarks = 1
2 Set Theory = 3
3 Mapping = 8
4 A(S)(The Set of 1-1 Mappings of S onto Itself) = 18
5 The Integers = 25
6 Mathematical Induction = 32
7 Complex Numbers = 37
CHAPTER 2 Groups = 47
1 Definitions ions and Examples of Groups = 47
2 Some Simple Remarks = 57
3 Subgroups = 59
4 Lagrange's Theorem = 66
5 Homomorphisms and Normal Subgroups = 79
6 Factor Groups = 92
7 The Homomorphism Theorems = 100
8 Cauchy's Theorem = 105
9 Direct Products = 110
10 Finite Abelian Groups(Optional) = 115
11 Conjugacy and Sylow's Theorem(Optional) = 120
CHAPTER 3 The Symmetric Group = 129
1 Preliminaries = 129
2 Cycle Decomposition = 133
3 Odd and Even Permutations = 140
CHAPTER 4 Ring Theory = 147
1 Definitions and Examples = 147
2 Some Simple Results = 161
3 Ideals, Homomorphisms, and Quotient Rings = 164
4 Maximal Ideals = 174
5 Polynomial Rings = 178
6 Polynomials over the Rationals = 194
7 Field of Quotients of an Integral Domain = 202
CHAPTER 5 Fields = 207
1 Examples of Fields = 208
2 A Brief Excursion into Vector Spaces = 212
3 Field Extensions = 225
4 Finite Extensions = 234
5 Constructibility = 237
6 Roots of Polynomials = 245
CHAPTER 6 Special Topic(Optional) = 255
1 The Simplicity of An = 256
2 Finite Fields Ⅰ = 263
3 Finite Fields Ⅱ : Existence = 266
4 Finite Fields Ⅲ : Uniqueness = 270
5 Cyclotomic Polynomials = 272
6 Liouville's Criterion = 281
7 The Irrationality of π = 285
Index = 289
