CONTENTS
Preface = ⅴ
Leirfaden = xiii
Notational conventions = xiv
Chapter 1. Introduction. = 1
1. Valuations = 1
2. Remarks = 4
3. An application = 6
Chapter 2. General properties = 12
1. Definitions and basics = 12
2. Valuations on the rationals = 16
3. Independence of valuations = 18
4. Completion = 23
5. Formal series and a theorem of Eisenstein = 26
Chapter 3. Archimedean valuations = 33
1. Introduction = 33
2. Some lemmas = 34
3. Completion of proof = 38
Chapter 4. Non archimedean valuations. Simple properties = 41
1. Definitions and basics = 41
2. An application to finite groups of rational matrices = 46
3. Hensel's lemma = 49
3. bis. Application to diophantine equations = 53
4. Elementary analysis = 59
5. A useful expansion = 64
6. An application to recurrent sequences = 67
Chapter 5. Embedding theorem = 82
1. Introduction = 82
2. Three lemmas = 82
3. Proof of theorem = 84
4. Application. A theorem of Selberg = 87
5. Application. The theorem of Mahler and Lech = 88
Chapter 6. Transcendental extensions. Factorization = 92
1. Introduction = 92
2. Gauss' lemma and Eisenstein irreducibility = 95
3. Newton polygon = 98
4. Factorization of pure polynomials = 105
5. "Weierstrass" preparation theorem = 107
Chapter 7. Algebraic extensions (complete fields) = 114
1. Introduction = 114
2. Uniqueness = 115
3. Existence and corollaries = 118
4. Residue class fields = 120
5. Ramification = 123
6. Discriminants = 128
7. Completely ramified extensions = 133
8. Action of galois = 134
Chapter 8. p-adic fields = 144
1. Introduction = 144
2. Unramified extensions = 147
3. Non-completeness of Q ? p = 149
4. "Kronecker-Weber" theorem = 151
Chapter 9. Algebraic extensions (incomplete fields) = 165
1. Introduction 165
2. Proof of theorem : norm and trace = 167
3. Integers and discriminants = 170
4. Application to cyclotomic fields = 174
5. Action of galois = 176
6. Application. Quadratic reciprocity = 178
Chapter 10. Algebraic number fields = 189
1. Introduction = 189
2. Product formula = 190
3. Algebraic integers = 191
4. Strong approximation theorem = 196
5. Divisors. Relation to ideal theory = 197
6. Existence theorems = 203
7. Finiteness of the class number = 208
8. The unit group = 211
9. Application to diophantine equations. Rational solutions = 220
10. Application to diophantine equations. Integral solutions = 222
10. bis. Application to diophantine equations. Integral solutions. (contd) = 228
11. The discriminant = 231
12. The Kronecker-Weber theorem = 235
13. Statistics of prime decomposition = 237
Chapter 11. Diophantine equations = 250
1. Introduction = 250
2. Basse Principle for ternary quadratics = 252
3. Curves of genus 1. Generalities = 257
4. Curves of genus 1. A special case = 261
Chapter 12. Advanced analysis = 280
1. Introduction = 280
2. Elementary functions = 280
3. Analytic continuation = 285
4. Measure on Zp = 288
5. The zeta function = 291
6. L-functions = 296
7. Mahler's expansion = 306
Chapter 13. A theorem of Borel and Dwork = 313
1. Introduction = 313
2. Some lemmas = 314
3. Proof = 317
Appendix A. Resultants and discriminants = 320
Appendix B. Norms, traces and characteristic polynomials = 325
Appendix C. Minkowski's convex body theorem = 331
Appendix D. Solution of equations in finite fields = 337
Appendix E. Zeta and L-functions at negative integers = 347
Appendix F. Calculation of exponentials = 350
References = 352
Index = 358
