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PREFACE

CONTENTS

LEITFADEN

NOTATIONAL CONVENTIONS

CHAPTER ONE: INTRODUCTION

1 VALUATIONS

2 REMARKS

3 AN APPLICATION

CHAPTER TWO: GENERAL PROPERTIES

1 DEFINITIONS AND BASICS

2 VALUATIONS ON THE RATIONALS

3 INDEPENDENCE OF VALUATIONS

4 COMPLETENESS

5 FORMAL SERIES AND A THEOREM OF EISENSTEIN

CHAPTER THREE: ARCHIMEDEAN VALUATIONS

1 INTRODUCTION

2 SOME LEMMAS

3 COMPLETION OF PROOF

CHAPTER FOUR: NON–ARCHIMEDEAN VALUATIONS. SIMPLE PROPERTIES

1 DEFINITIONS AND BASICS

2 AN APPLICATION TO FINITE GROUPS OF RATIONAL MATRICES

3 HENSEL'S LEMMA

4 ELEMENTARY ANALYSIS

5 A USEFUL EXPANSION

6 AN APPLICATION TO RECURRENT SEQUENCES

CHAPTER FIVE: EMBEDDING THEOREM

1 INTRODUCTION

2 THREE LEMMAS

3 PROOF OF THEOREM

4 APPLICATION. A THEOREM OF SELBERG

5 APPLICATION. THE THEOREM OF MAHLER AND LECH

CHAPTER SIX: TRANSCENDENTAL EXTENSIONS. FACTORIZATION

1 INTRODUCTION

2 GAUSS' LEMMA AND EISENSTEIN IRREDUCIBILITY

3 NEWTON POLYGON

4 FACTORIZATION OF PURE POLYNOMIALS

5 WEIERSTRASS PREPARATION THEOREM

CHAPTER SEVEN: ALGEBRAIC EXTENSIONS (COMPLETE FIELDS)

1 INTRODUCTION

2 UNIQUENESS

3 EXISTENCE

4 RESIDUE CLASS FIELDS

5 RAMIFICATION

6 DISCRIMINANTS

7 COMPLETELY RAMIFIED EXTENSIONS

8 ACTION OF GALOIS

CHAPTER EIGHT: P-ADIC FIELDS

1 INTRODUCTION

2 UNRAMIFIED EXTENSIONS

3 NON-COMPLETENESS OF Qp

4 "KRONECKER-WEBER" THEOREM

CHAPTER NINE: ALGEBRAIC EXTENSIONS (INCOMPLETE FIELDS)

1 INTRODUCTION

2 PROOF OF THEOREM AND COROLLARIES

3 INTEGERS AND DISCRIMINANTS

4 APPLICATION TO CYCLOTOMIC FIELDS

5 ACTION OF GALOIS

6 APPLICATION. QUADRATIC RECIPROCITY

CHAPTER TEN: ALGEBRAIC NUMBER FIELDS

1 INTRODUCTION

2 PRODUCT FORMULA

3 ALGEBRAIC INTEGERS

4 STRONG APPROXIMATION THEOREM

5 DIVISORS. RELATION TO IDEAL THEORY

6 EXISTENCE THEOREMS

7 FINITENESS OF THE CLASS NUMBER

8 THE UNIT GROUP

9 APPLICATION TO DIOPHANTINE EQUATIONS. RATIONAL SOLUTIONS

10 APPLICATION TO DIOPHANTINE EQUATIONS. INTEGRAL SOLUTIONS

11 THE DISCRIMINANT

12 THE KRONECKER-WEBER THEOREM

13 STATISTICS OF PRIME DECOMPOSITION

CHAPTER ELEVEN: DIOPHANTINE EQUATIONS

I INTRODUCTION

2 HASSE PRINCIPLE FOR TERNARY QUADRATICS

3 CURVES OF GENUS 1. GENERALITIES

4 CURVES OF GENUS 1. A SPECIAL CASE

CHAPTER TWELVE: ADVANCED ANALYSIS

1 INTRODUCTION

2 ELEMENTARY FUNCTIONS

3 ANALYTIC CONTINUATION

4 MEASURE ON Zp

5 THE ZETA FUNCTION

6 L-FUNCTIONS

7 MAHLER'S EXPANSION

CHAPTER THIRTEEN: A THEOREM OF BOREL AND DWORK

1 INTRODUCTION

2 SOME LEMMAS

3 PROOF

APPENDIX A: RESULTANTS AND DISCRIMINANTS

APPENDIX B: NORMS, TRACES AND CHARACTERISTIC POLYNOMIALS

APPENDIX C: MINKOWSKI'S CONVEX BODY THEOREM

APPENDIX D: SOLUTION OF EQUATIONS IN FINITE FIELDS

APPENDIX E: ZETA AND L-FUNCTIONS AT NEGATIVE INTEGERS

APPENDIX F: CALCULATION OF EXPONENTIALS

REFERENCES

INDEX