**Mathematical Surveys and Monographs**

Volume: 6;
1951;
188 pp;
Softcover

MSC: Primary 14;

Print ISBN: 978-0-8218-1506-9

Product Code: SURV/6

List Price: $47.00

AMS Member Price: $37.60

MAA member Price: $42.30

**Electronic ISBN: 978-1-4704-1233-3
Product Code: SURV/6.E**

List Price: $47.00

AMS Member Price: $37.60

MAA member Price: $42.30

# Introduction to the Theory of Algebraic Functions of One Variable

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*C. C. Chevalley*

This classical book, written by a famous French
mathematician in the early 1950s, presents an approach to algebraic
geometry of curves treated as the theory of algebraic functions on the
curve. Among other advantages of such an approach, it allowed the
author to consider curves over an arbitrary ground field. Among topics
discussed in the book are the theory of divisors on a curve, the
Riemann–Roch theorem, \(p\)-adic completion, extensions
of the fields of functions (covering theory) and of the fields of
constants, and the theory of differentials on a curve. The last
chapter, which is more analytic in flavor, treats the theory of
Riemann surfaces.

Prerequisites for reading are minimal and include only an advanced
undergraduate algebra course.

#### Table of Contents

# Table of Contents

## Introduction to the Theory of Algebraic Functions of One Variable

- TABLE OF CONTENTS vii8 free
- INTRODUCTION ix10 free
- NOTATIONS FREQUENTLY USED xiii14 free
- CHAPTER I. PLACES AND DIVISORS 116 free
- CHAPTER II. THE THEOREM OF RIEMANN-ROCH 2035
- CHAPTER III. THE d-ADIC COMPLETIONS 3954
- CHAPTER IV. EXTENSIONS OF FIELDS OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE 5166
- 1. The relative degree and the ramification index 5166
- 2. The case of normal algebraic extensions 5368
- 3. Integral bases 5469
- 4. Kronecker products of commutative algebras 5772
- 5. Extension of the d-adic completion 5974
- 6. The Puiseux expansions 6479
- 7. Norm and conorm; trace and cotrace 6580
- 8. The different 6984
- 9. Structure of hyperelliptic fields 7489

- CHAPTER V. EXTENSIONS OF THE FIELD OF CONSTANTS 7994
- CHAPTER VI. EXACT DIFFERENTIALS 101116
- 1. The differential dx in K(x) 101116
- 2. Trace and cotrace of differentials 103118
- 3. The differential dx in an arbitrary fields 108123
- 4. Derivations of fields 111126
- 5. Derivations and differentials 116131
- 6. Extension of the notion of cotrace 118133
- 7. Derivations of the field of constants 125140
- 8. Differentials of the second kind 127142

- CHAPTER VII. THE RIEMANN SURFACE 133148
- 1. Definition of the Riemann surface 133148
- 2. Meromorphic functions on the Riemann surface 136151
- 3. On singular homology theory 141156
- 4. Periods of differentials 145160
- 5. The bilinear function j(ω, ω') 153168
- 6. Definition of the intersection numbers 156171
- 7. Geometric lemmas 162177
- 8. The homology groups of the Riemann surface 166181
- 9. The theorem of Abel 172187
- 10. Fields of genus one 176191
- 11. The Riemann surface as an analytic manifold 178193
- 12. The bilinear inequalities of Riemann 182197

- INDEX 187202