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

v

vi *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 130

*Contents* 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*[*λ*1*, . . . , 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 220

viii *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**

Introduction 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 309

*Contents* 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