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
