Table of Contents
Preface v
1 Preliminaries 1
1.1 Basic Ideas of Set Theory 2
1.2 Functions 7
1.3 Equivalence Relations and Partitions 11
1.4 A Note on Natural Numbers 14
Review Exercises 16
2 Algebraic Structure of Numbers 17
2.1 The Set of Integers 18
2.2 Congruences of Integers 21
2.3 Rational Numbers 28
Review Exercises 33
3 Basic Notions of Groups 35
3.1 Definitions and Examples 36
3.2 Basic Properties 41
3.3 Subgroups 45
3.4 Generating Sets 48
Review Exercises 51
4 Cyclic Groups 53
4.1 Cyclic Groups 54
4.2 Subgroups of Cyclic Groups 57
Review Exercises 63
5 Permutation Groups 65
5.1 Symmetric Groups 66
5.2 Dihedral Groups 71
5.3 Alternating Groups 76
Review Exercises 79
6 Counting Theorems 81
6.1 Lagrange's Theorem 82
6.2 Conjugacy Classes of a Group 87
Review Exercises 93
7 Group Homomorphisms 95
7.1 Examples and Basic Properties 96
7.2 Isomorphisms 99
7.3 Cayley's Theorem 105
Review Exercises 108
8 The Quotient Group 109
8.1 Normal Subgroups 110
8.2 Quotient Groups 114
8.3 Fundamental Theorem of Group Homomorphisms 119
Review Exercises 125
9 Finite Abelian Groups 127
9.1 Direct Products of Groups 128
9.2 Cauchy's Theorem 133
9.3 Structure Theorem of Finite Abelian Groups 137
Review Exercises 142
10 Sylow Theorems and Applications 143
10.1 Group Actions 144
10.2 Sylow Theorems 151
Review Exercises 157
11 Introduction to Group Presentations 159
11.1 Free Groups and Free Abelian Groups 160
11.2 Generators and Relations 165
11.3 Classification of Finite Groups of Small Orders 170
Review Exercises 175
12 Types of Rings 177
12.1 Definitions and Examples 178
12.2 Matrix Rings 185
Review Exercises 191
13 Ideals and Quotient Rings 193
13.1 Ideals 194
13.2 Quotient Rings 198
Review Exercises 203
14 Ring Homomorphisms 205
14.1 Ring Homomorphisms 206
14.2 Direct Products of Rings 211
14.3 The Quotient Field of an Integral Domain 216
Review Exercises 222
15 Polynomial Rings 223
15.1 Polynomial Rings in the Indeterminates 224
15.2 Properties of the Polynomial Rings of One Variable 228
15.3 Principal Ideal Domains and Euclidean Domains 233
Review Exercises 237
16 Factorization 239
16.1 Irreducible and Prime Elements 240
16.2 Unique Factorization Domains 245
16.3 Polynomial Extensions of Factorial Domains 253
Review Exercises 259
17 Vector Spaces Over an Arbitrary Field 261
17.1 A Brief Review on Vector Spaces 262
17.2 A Brief Review on Linear Transformations 266
Review Exercises 272
18 Field Extensions 273
18.1 Algebraic or Transcendental? 274
18.2 Finite and Algebraic Extensions 278
18.3 Construction with Straightedge and Compass 284
Review Exercises 294
19 All About Roots 295
19.1 Zeros of Polynomials 296
19.2 Uniqueness of Splitting Fields 299
19.3 Algebraically Closed Fields 303
19.4 Multiplicity of Roots 305
19.5 Finite Fields 309
Review Exercises 314
20 Galois Pairing 315
20.1 Galois Groups 316
20.2 The Fixed Subfields of a Galois Group 321
20.3 Fundamental Theorem of Galois Pairing 326
Review Exercises 331
21 Applications of the Galois Pairing 333
21.1 Fields of Invariants 334
21.2 Solvable Groups 338
21.3 Insolvability of the Quintic 345
Review Exercises 350
Index 351
