Introduction to the Theory of Algebraic Functions of One VariableAmerican Mathematical Soc., 31 déc. 1951 - 188 pages 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 des matières
PLACES AND DIVISORS _ | 1 |
c | 6 |
The order function The degree of a place | 9 |
The theorem of independence | 11 |
Divisors | 13 |
The divisor of a function | 15 |
THE THEOREM 0F RIEMANNROCH | 20 |
Fields of genus zero | 23 |
EXTENSIONS OF THE FIELD or CONSTANTS _ | 79 |
Relatively algebraically closed subfields | 82 |
Commutative algebras | 85 |
Definition of the extended field | 88 |
The effect on a place v | 92 |
The effect on the genus _ | 96 |
EXACT DIFFERENTIALS | 101 |
Trace and cotrace of differentials _ | 103 |
Fields of genus one _ _ | 24 |
Repartitions | 25 |
Difierentials | 28 |
The canonical class _ | 31 |
The local components of a differential | 33 |
Fields of elliptic functions | 34 |
THE pADIC COMPLETIONS | 39 |
Hensels lemma | 43 |
Structure of padic completions | 45 |
Generalization of the notion of repartition | 46 |
Residues of a differential | 48 |
EXTENSIONS or FIELDS or ALGEBRAIC FUNCTIONS or ONE VARIABLE | 51 |
The case of normal algebraic extensions | 53 |
Integral bases | 54 |
Kronecker products of commutative algebras v | 57 |
Extension of the padic completion | 59 |
The Puiseux expansions | 64 |
Norm and conorm trace and cotrace | 65 |
The different _ | 69 |
Structure of hyperelliptic fields | 75 |
in an arbitrary field | 108 |
Derivations of fields | 111 |
Derivations and differentials | 116 |
Extension of the notion of cotrace | 119 |
Derivations of the field of constants _ _ | 125 |
Differentials of the second kind | 127 |
THE RIEMANN SURFACE | 133 |
Meromorphic functions on the Riemann surface | 136 |
On singular homology theory | 141 |
Periods of differentials | 145 |
The bilinear function jw w | 153 |
Definition of the intersection numbers | 156 |
Geometric lemmas _ | 162 |
The homology groups of the Riemann surface | 166 |
The theorem of Abel _ | 173 |
Fields of genus one | 177 |
The Riemann surface as an analytic manifold | 178 |
The bilinear inequalities of Riemann | 183 |
| 187 | |
Autres éditions - Tout afficher
Introduction to the Theory of Algebraic Functions of One Variable Claude Chevalley Aucun aperçu disponible - 1951 |
Introduction to the Theory of Algebraic Functions of One Variable Claude Chevalley Aucun aperçu disponible - 1951 |
Introduction to the Theory of Algebraic Functions of One Variable Claude Chevalley Aucun aperçu disponible - 1951 |
Expressions et termes fréquents
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