TABLE OF CONTENTS INTRODUCTION be TABL E O F CONTENTS vii NOTATIONS FREQUENTLY USE D xiii Chapte r I. PLACES AND DIVISOR S 1 1. Fields of algebraic functions of one variable 1 2. Places 1 3. Places of the field K(x) 2 4. Existence of places 6 5. The order function. The degree of a place 9 6. The theorem of independence 11 7. Divisors 13 8. The divisor of a function 15 Chapte r II . T H E THEORE M OF RIEMANN-ROC H 20 1. The genus 20 2. Fields of genus zero 23 3. Fields of genus one 24 4. Repartitions 25 5. Differentials 28 6. The canonical class 31 7. The local components of a differential 33 8. Fields of elliptic functions 34 Chapte r III . T H E P-ADIC COMPLETIONS 39 1. Definition of the p-adic completion 39 2. HensePs lemma 43 3. Structure of p-adic completions 44 4. Generalization of the notion of repartition 46 5. Residues of a differential 48 Chapte r IV. EXTENSION S OF FIELD S OF ALGEBRAIC FUNCTIONS OF O N E VARIABLE 51 1. The relative degree and the ramification index 51 2. The case of normal algebraic extensions 53 3. Integral bases 54 4. Kronecker products of commutative algebras 57 5. Extension of the p-adic completion 59 6. The Puiseux expansions 64 7. Norm and conorm trace and cotrace 65 8. The different 69 9. Structure of hyperelliptic fields 74 Chapte r V. EXTENSION S O F THE FIEL D OF CONSTANTS 79 1. Separable transcendental extensions 79 2. Relatively algebraically closed subfields 82 3. Commutative algebras 85 4. Definition of the extended field 88 5. The effect on a place 92 6. The effect on the genus 96 Chapte r VI . EXACT DIFFERENTIAL S 101 1. The differential dx in K{x) 101 2. Trace and cotrace of differentials 103 3. The differential dx in an arbitrary field 108 vii
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