Algebra (eBook)

Front Cover 1
Algebra 4
Copyright Page 5
Contents 6
Preface 8
List of Symbols 10
Introduction 14
Chapter I. Groups 16
1. Groups, Subgroups, and Homomorphisms 16
2. Permutation Groups 27
3. The Symmetric and Alternating Groups 31
4. The Sylow Theorems 34
5. Solvable Groups, Normal and Subnormal Series 37
6. Products 42
7. Nilpotent Groups 44
8. Finite Abelian Groups 47
9. Free Groups 48
10. Generators and Relations 52
11. Some Finite Groups Classified 55
12. Further Exercises 56
Chapter II. Rings 62
1. Preliminaries: Ideals and Homomorphisms 62
2. The Field of Fractions of an Integral Domain 67
3. Polynomials 69
4. Polynomials in Several Indeterminates 73
5. Divisibility and Factorization 76
6. The Chinese Remainder Theorem 86
7. The Hilbert Basis Theorem 88
8. Further Exercises 90
Chapter III. Fields and Galois Theory 94
1. Field Extensions 94
2. The Fundamental Theorem of Galois Theory 101
3. Normality and Separability 105
4. The Galois Theory of Equations 111
5. Symmetric Functions 117
6. Geometrical Constructions 124
7. Norm and Trace 130
8. Further Exercises 134
Chapter IV. Modules 140
1. Preliminaries 140
2. Direct Sums, Free Modules 144
3. Finitely Generated Modules over a PID 149
4. Applications to Linear Algebra 155
5. Computations 160
6. Tensor Products 173
7. Further Exercises 179
Chapter V. Structure of Rings and Algebras 187
1. Preliminaries 187
2. The Jacobson Radical 191
3. The Density Theorem 195
4. Artinian Rings 198
5. Further Exercises 204
Chapter VI. Further Topics 209
1. Infinite Abelian Groups 209
2. PÓlya Redfield Enumeration 215
3. PSL(V) 225
4. Integral Dependence and Dedekind Domains 232
5. Transcendental Field Extensions 245
6. Valuations and p-adic Numbers 253
7. Real Fields and Sturm’s Theorem 267
8. Representations and Characters of Finite Groups 275
9. Some Galois Groups 290
Appendix: Zorn’s Lemma 304
References 306
Index 308
Pure and Applied Mathematics 315