Algebra

Preface xi
Chapter 1. Arithmetic 1
1. Integers 1
1.1. Basic Properties 1
1.2. Induction 2
1.3. Division Algorithm 4
1.4. Divisors 5
1.5. Fundamental Theorem of Arithmetic 8
1.6. Exercises 1.1–1.28 9
2. Modular Arithmetic 13
2.1. Congruence 13
2.2. Congruence Classes 14
2.3. Arithmetic 15
2.4. Structure 16
2.5. Applications 20
2.6. Equivalence Relations 24
2.7. Exercises 1.29–1.72 26
Chapter 2. Groups 33
1. Definitions and Examples 33
1.1. Binary Operations 33
1.2. Definition of a Group 34
1.3. Examples 35
1.4. Exercises 2.1–2.29 41
2. Basic Properties and Order 46
2.1. Exercises 2.30–2.52 48
3. Subgroups and Direct Products 51
3.1. Subgroups 51
3.2. Direct Products 54
3.3. Exercises 2.53–2.91 54
4. Morphisms 59
4.1. Introduction 59
4.2. Definitions and Examples 60
4.3. Basic Properties 62
4.4. Exercises 2.92–2.118 63
viiviii CONTENTS
5. Quotients 66
5.1. Definitions 66
5.2. Lagrange’s Theorem 68
5.3. Normality 69
5.4. Direct Products and the Correspondence Theorem 72
5.5. First Isomorphism Theorem 73
5.6. Exercises 2.119–2.169 74
6. Fundamental Theorem of Finite Abelian Groups 81
6.1. Exercises 2.170–2.188 85
7. The Symmetric Group 89
7.1. Cayley’s Theorem 89
7.2. Cyclic Decomposition 90
7.3. Conjugacy Classes 93
7.4. Parity and the Alternating Subgroup 95
7.5. Exercises 2.189–2.229 97
8. Group Actions 103
8.1. Exercises 2.230–2.248 109
9. Sylow Theorems 113
9.1. Exercises 2.249–2.281 118
10. Simple Groups and Composition Series 123
10.1. Simple Groups 123
10.2. Composition Series 126
10.3. Exercises 2.282–2.301 128
Chapter 3. Rings 131
1. Examples and Basic Properties 131
1.1. Definition 131
1.2. Examples 132
1.3. Basic Properties 132
1.4. Subrings 134
1.5. Direct Products 135
1.6. Exercises 3.1–3.39 136
2. Morphisms and Quotients 141
2.1. Morphisms 141
2.2. Ideals 143
2.3. Quotients 144
2.4. Isomorphism Theorem 147
2.5. Exercises 3.40–3.86 147
3. Polynomials and Roots 155
3.1. Division Algorithm 157
3.2. Roots 158
3.3. Rational Root Test 160
3.4. Fundamental Theorem of Algebra 161
3.5. Exercises 3.87–3.129 162CONTENTS ix
4. Polynomials and Irreducibility 169
4.1. Irreducibility 169
4.2. Polynomials over a Field 169
4.3. Exercises 3.130–3.150 172
5. Factorization 174
5.1. Unique Factorization 175
5.2. Quotient Fields 178
5.3. Unique Factorization in Polynomial Rings 179
5.4. Exercises 3.151–3.172 181
6. Principal Ideal and Euclidean Domains 183
6.1. Principal Ideal Domains 183
6.2. Euclidean Domains 184
6.3. Gaussian Integers 185
6.4. Exercises 3.173–3.201 186
Chapter 4. Field Theory 193
1. Finite and Algebraic Extensions 193
1.1. Vector Spaces 193
1.2. Finite and Algebraic Extensions 196
1.3. Exercises 4.1–4.34 200
2. Splitting Fields 204
2.1. Splitting Fields 204
2.2. Algebraic Closures 208
2.3. Exercises 4.35–4.62 211
3. Finite Fields 215
3.1. Exercises 4.63–4.81 217
4. Galois Theory 219
4.1. Galois Groups 219
4.2. Separability 222
4.3. Galois Correspondence 223
4.4. Exercises 4.82—4.135 229
5. Famous Impossibilities 236
5.1. Compass and Straightedge Constructions 236
5.2. Solvability of Polynomials 239
5.3. Exercises 4.136–4.162 243
6. Cyclotomic Fields 245
6.1. Exercises 4.163–4.179 247
Index 251