數論第1卷

《數論第1卷》是2019年世界圖書出版公司出版的圖書，作者是[法] H.科恩。

基本介紹

• 中文名：數論第1卷
• 作者：[法] H.科恩
• 出版社：世界圖書出版公司
• 出版時間：2019年2月1日
• ISBN：9787519255299

內容簡介

《數論》分為2卷，是Springer“數學研究生教材”叢書之239和240卷，是一套面向研究生的數論教程，主旨是全面介紹丟番圖方程的解，包括多項式方程、有理數和代數數論等，其中特彆強調了算術代數幾何的現代理論。全書各章共有530例習題，部分習題有提示。

圖書目錄

Volume I

Preface

1. Introduction to Diophantine Equations

1.1 Introduction

1.1.1 Examples of Diophantine Problems

1.1.2 Local Methods

1.1.3 Dimensions

1.2 Exercises for Chapter 1

Part I. Tools

2. Abelian Groups, Lattices, and Finite Fields

2.1 Finitely Generated Abelian Groups

2.1.1 Basic Results

2.1.2 Description of Subgroups

2.1.3 Characters of Finite Abelian Groups

2.1.4 The Groups (Z/mZ)*

2.1.5 Dirichlet Characters

2.1.6 Gauss Sums

2.2 The Quadratic Reciprocity Law

2.2.1 The Basic Quadratic Reciprocity Law

2.2.2 Consequences of the Basic Quadratic Reciprocity Law

2.2.3 Gauss's Lemma and Quadratic Reciprocity

2.2.4 Real Primitive Characters

2.2.5 The Sign of the Quadratic Gauss Sum

2.3 Lattices and the Geometry of Numbers

2.3.1 Definitions

2.3.2 Hermite's Inequality

2.3.3 LLL-Reduced Bases

2.3.4 The LLL Algorithms

2.3.5 Approximation of Linear Forms

2.3.6 Minkowski's Convex Body Theorem

2.4 Basic Properties of Finite Fields

2.4.1 General Properties of Finite Fields

2.4.2 Galois Theory of Finite Fields

2.4.3 Polynomials over Finite Fields

2.5 Bounds for the Number of Solutions in Finite Fields

2.5.1 The Chevalley-Warning Theorem

2.5.2 Gauss Sums for Finite Fields

2.5.3 Jacobi Sums for Finite Fields

2.5.4 The Jacobi Sums J(x1,x2)

2.5.5 The Number of Solutions of Diagonal Equations

2.5.6 The Well Bounds

2.5.7 The Weil Conjectures (Deligne's Theorem)

2.6 Exercises for Chapter 2

3. Basic Algebraic Number Theory

3.1 Field-Theoretic Algebraic Number Theory

3.1.1 Galois Theory

3.1.2 Number Fields

3.1.3 Examples

3.1.4 Characteristic Polynomial, Norm, Trace

3.1.5 Noether's Lemma

3.1.6 The Basic Theorem of Kummer Theory

3.1.7 Examples of the Use of Kummer Theory

3.1.8 Artin-Schreier Theory

3.2 The Normal Basis Theorem

3.2.1 Linear Independence and Hilbert's Theorem 90

3.2.2 The Normal Basis Theorem in the Cyclic Case

3.2.3 Additive Polynomials

3.2.4 Algebraic Independence of Homomorphisms

3.2.5 The Normal Basis Theorem

3.3 Ring-Theoretic Algebraic Number Theory

3.3.1 Gauss's Lemma on Polynomials

3.3.2 Algebraic Integers

3.3.3 Ring of Integers and Discriminant

3.3.4 Ideals and Units

3.3.5 Decomposition of Primes and Ramification

3.3.6 Galois Properties of Prime Decomposition

3.4 Quadratic Fields

3.4.1 Field-Theoretic and Basic Ring-Theoretic Properties

3.4.2 Results and Conjectures on Class and Unit Groups

3.5 Cyclotomic Fields

3.5.1 Cyclotomic Polynomials

3.5.2 Field-Theoretic Properties of Q(Sn)

3.5.3 Ring-Theoretic Properties

3.5.4 The Totally Real Subfield of Q(Spk )

……

4. p-adic Fields

5. Quadratic Forms and Local-Global Principles

Part II. Diophantine Equations

6. Some Diophantine Equations

7. Elliptic Curves

8. Diophantine Aspects of Elliptic Curves

Bibliography

Index of Notation

Index of Names

General Index

作者簡介

這套經典研究生教材的作者作者Henri Cohen(H.科恩，法國)， 是法國Institue de Mathématiques de Bordeaux的教授。