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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