Graduate algebra: commutative view

Introduction xi
List of symbols xv

Chapter 0. Introduction and Prerequisites 1
Groups 2
Rings 6
Polynomials 9
Structure theories 12
Vector spaces and linear algebra 13
Bilinear forms and inner products 15
Appendix 0A: Quadratic Forms 18
Appendix 0B: Ordered Monoids 23
Exercises – Chapter 0 25
Appendix 0A 28
Appendix 0B 31

Part I. Modules

Chapter 1. Introduction to Modules and their Structure Theory 35
Maps of modules 38
The lattice of submodules of a module 42
Appendix 1A: Categories 44
vvi Contents

Chapter 2. Finitely Generated Modules 51
Cyclic modules 51
Generating sets 52
Direct sums of two modules 53
The direct sum of any set of modules 54
Bases and free modules 56
Matrices over commutative rings 58
Torsion 61
The structure of finitely generated modules over a PID 62
The theory of a single linear transformation 71
Application to Abelian groups 77
Appendix 2A: Arithmetic Lattices 77

Chapter 3. Simple Modules and Composition Series 81
Simple modules 81
Composition series 82
A group-theoretic version of composition series 87
Exercises — Part I 89
Chapter 1 89
Appendix 1A 90
Chapter 2 94
Chapter 3 96


Part II. Affine Algebras and Noetherian Rings

Introduction to Part II 99

Chapter 4. Galois Theory of Fields 101
Field extensions 102
Adjoining roots of a polynomial 108
Separable polynomials and separable elements 114
The Galois group 117
Galois extensions 119
Application: Finite fields 126
The Galois closure and intermediate subfields 129
Chains of subfields 130Contents vii
Application: Algebraically closed fields and the
algebraic closure 133
Constructibility of numbers 135
Solvability of polynomials by radicals 136
Supplement: Trace and norm 141
Appendix 4A: Generic Methods in Field Theory:
Transcendental Extensions 146
Transcendental field extensions 146
Appendix 4B: Computational Methods 150
The resultant of two polynomials 151
Appendix 4C: Formally Real Fields 155

Chapter 5. Algebras and Affine Fields 157
Affine algebras 161
The structure of affine fields – Main Theorem A 161
Integral extensions 165

Chapter 6. Transcendence Degree and the Krull Dimension
of a Ring 171
Abstract dependence 172
Noether normalization 178
Digression: Cancellation 180
Maximal ideals of polynomial rings 180
Prime ideals and Krull dimension 181
Lifting prime ideals to related rings 184
Main Theorem B 188
Supplement: Integral closure and normal domains 189
Appendix 6A: The automorphisms of F[λ1,...,λn] 194
Appendix 6B: Derivations of algebras 197

Chapter 7. Modules and Rings Satisfying Chain Conditions 207
Noetherian and Artinian modules (ACC and DCC) 207
Noetherian rings and Artinian rings 210
Supplement: Automorphisms, invariants, and Hilbert’s
fourteenth problem 214
Supplement: Graded and filtered algebras 217
Appendix 7A: Gr¨obner bases 220viii Contents

Chapter 8. Localization and the Prime Spectrum 225
Localization 225
Localizing the prime spectrum 230
Localization to local rings 232
Localization to semilocal rings 235

Chapter 9. The Krull Dimension Theory of Commutative
Noetherian Rings 237
Prime ideals of Artinian and Noetherian rings 238
The Principal Ideal Theorem and its generalization 240
Supplement: Catenarity of affine algebras 242
Reduced rings and radical ideals 243
Exercises – Part II 247
Chapter 4 247
Appendix 4A 257
Appendix 4B 258
Appendix 4C 262
Chapter 5 264
Chapter 6 264
Appendix 6B 268
Chapter 7 274
Appendix 7A 276
Chapter 8 277
Chapter 9 280

Part III. Applications to Geometry and Number Theory
I
ntroduction to Part III 287

Chapter 10. The Algebraic Foundations of Geometry 289
Affine algebraic sets 290
Hilbert’s Nullstellensatz 293
Affine varieties 294
Affine “schemes” 298
Projective varieties and graded algebras 303
Varieties and their coordinate algebras 308
Appendix 10A. Singular points and tangents 309Contents ix

Chapter 11. Applications to Algebraic Geometry over the Rationals–
Diophantine Equations and Elliptic Curves 313
Curves 315
Cubic curves 318
Elliptic curves 322
Reduction modulo p 337

Chapter 12. Absolute Values and Valuation Rings 339
Absolute values 340
Valuations 346
Completions 351
Extensions of absolute values 356
Supplement: Valuation rings and the integral closure 361
The ramification index and residue field 363
Local fields 369
Appendix 12A: Dedekind Domains and Class Field Theory 371
The ring-theoretic structure of Dedekind domains 371
The class group and class number 378
Exercises – Part III 387
Chapter 10 387
Appendix 10A 390
Chapter 11 391
Chapter 12 397
Appendix 12A 404

List of major results 413

Bibliography 427

Index 431