関数体における数論<br>Number Theory in Function Fields (Graduate Texts in Mathematics) 〈Vol. 210〉

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関数体における数論
Number Theory in Function Fields (Graduate Texts in Mathematics) 〈Vol. 210〉

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  • 製本 Hardcover:ハードカバー版/ページ数 310 p.
  • 商品コード 9780387953359

基本説明

Contents: Polynomials over Finite Fields; Primes, Arithmetic Functions, and the Zeta Function; The Reciprocity Law; and more.

Full Description


Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.

Table of Contents

Preface                                            vii
Polynomials over Finite Fields 1 (10)
Exercises 7 (4)
Primes, Arithmetic Functions, and the Zeta 11 (12)
Function
Exercises 19 (4)
The Reciprocity Law 23 (10)
Exercises 30 (3)
Dirichlet L-Series and Primes in an 33 (12)
Arithmetic Progression
Exercises 43 (2)
Algebraic Function Fields and Global Function 45 (18)
Fields
Exercises 59 (4)
Weil Differentials and the Canonical Class 63 (14)
Exercises 75 (2)
Extensions of Function Fields, 77 (24)
Riemann-Hurwitz, and the ABC Theorem
Exercises 98 (3)
Constant Field Extensions 101(14)
Exercises 112(3)
Galois Extensions - Hecke and Artin L-Series 115(34)
Exercises 145(4)
Artin's Primitive Root Conjecture 149(20)
Exercises 166(3)
The Behavior of the Class Group in Constant 169(24)
Field Extensions
Exercises 190(3)
Cyclotomic Function Fields 193(26)
Exercises 216(3)
Drinfeld Modules: An Introduction 219(22)
Exercises 239(2)
S-Units, S-Class Group, and the Corresponding 241(16)
L-Functions
Exercises 256(1)
The Brumer-Stark Conjecture 257(26)
Exercises 278(5)
The Class Number Formulas in Quadratic and 283(22)
Cyclotomic Function Fields
Exercises 302(3)
Average Value Theorems in Function Fields 305(24)
Exercises 326(3)
Appendix: A Proof of the Function Field Riemann 329(12)
Hypothesis
Bibliography 341(12)
Author Index 353(2)
Subject Index 355