Preface ...................................................... vii
1 Groups ...................................................... 1
1.1 Introduction to Groups ................................. 1
1.2 Subgroups ............................................. 11
1.3 Homomorphisms of Groups ............................... 16
1.4 Group Structure ....................................... 22
1.5 The Sylow Theorems .................................... 31
1.6 Exercises ............................................. 35
2 Rings ...................................................... 41
2.1 Introduction to Rings ................................. 42
2.2 Polynomial Rings ...................................... 48
2.3 The Group of Units of a Ring .......................... 53
2.4 Ideals ................................................ 60
2.5 Quotient Rings and Ring Homomorphisms ................. 70
2.6 Localization .......................................... 78
2.7 Absolute Values and Completions ....................... 82
2.8 Exercises ............................................. 93
3 Modules ................................................... 101
3.1 Vector Spaces ........................................ 102
3.2 Modules .............................................. 109
3.3 Projective Modules ................................... 117
3.4 Tensor Products ...................................... 122
3.5 Algebras ............................................. 128
3.6 Discriminants ........................................ 133
3.7 Exercises ............................................ 140
4 Simple Algebraic Extension Fields ......................... 143
4.1 Simple Algebraic Extensions .......................... 144
4.2 Some Galois Theory ................................... 152
4.3 The Ring of Integers ................................. 158
4.4 The Noetherian Property of the Ring of Integers ...... 169
4.5 Dedekind Domains ..................................... 172
4.6 Unique Factorization of Ideals ....................... 178
4.7 Extensions of QP ..................................... 181
4.8 Exercises ............................................ 186
5 Finite Fields ............................................. 191
5.1 Invented Roots ....................................... 191
5.2 Finite Fields ........................................ 193
5.3 Linearly Recursive Sequences ......................... 200
5.4 Exercises ............................................ 212
Bibliography ................................................. 215
Index ........................................................ 217
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