Number Theory

xiiTable of Contents4.5.1 Strassmann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2684.5.2 Krasner Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . 2694.5.3 The Weierstrass Preparation Theorem . . . . . . . . . . . . . . 2724.5.4 Applications of Strassmann’s Theorem . . . . . . . . . . . . . . 2744.6 Exercises for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2785. Quadratic Forms and Local–Global Principles . . . . . . . . . . . . 2875.1 Basic Results on Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 2875.1.1 Basic Properties of Quadratic Modules . . . . . . . . . . . . . . 2885.1.2 Contiguous Bases and Witt’s Theorem . . . . . . . . . . . . . . 2905.1.3 Translations into Results on Quadratic Forms . . . . . . . . 2935.2 Quadratic Forms over Finite and Local Fields . . . . . . . . . . . . . . 2965.2.1 Quadratic Forms over Finite Fields . . . . . . . . . . . . . . . . . 2965.2.2 Definition of the Local Hilbert Symbol . . . . . . . . . . . . . . 2975.2.3 Main Properties of the Local Hilbert Symbol. . . . . . . . . 2985.2.4 Quadratic Forms over Q p . . . . . . . . . . . . . . . . . . . . . . . . . . 3025.3 Quadratic Forms over Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3055.3.1 Global Properties of the Hilbert Symbol . . . . . . . . . . . . . 3055.3.2 Statement of the Hasse–Minkowski Theorem . . . . . . . . . 3075.3.3 The Hasse–Minkowski Theorem for n 2 . . . . . . . . . . . 3085.3.4 The Hasse–Minkowski Theorem for n = 3 . . . . . . . . . . . 3095.3.5 The Hasse–Minkowski Theorem for n = 4 . . . . . . . . . . . 3105.3.6 The Hasse–Minkowski Theorem for n 5 . . . . . . . . . . . 3115.4 Consequences of the Hasse–Minkowski Theorem . . . . . . . . . . . . 3125.4.1 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3125.4.2 A Result of Davenport and Cassels . . . . . . . . . . . . . . . . . 3135.4.3 Universal Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . 3145.4.4 Sums of Squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3165.5 The Hasse Norm Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3205.6 The Hasse Principle for Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . 3235.6.1 A General Theorem on Powers . . . . . . . . . . . . . . . . . . . . . 3235.6.2 The Hasse Principle for Powers. . . . . . . . . . . . . . . . . . . . . 3265.7 Some Counterexamples to the Hasse Principle . . . . . . . . . . . . . . 3285.8 Exercises for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332Part II. Diophantine Equations6. Some Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.1.1 The Use of Finite Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . 3376.1.2 Local Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.1.3 Global Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3396.2 Diophantine Equations of Degree 1 . . . . . . . . . . . . . . . . . . . . . . . 3416.3 Diophantine Equations of Degree 2 . . . . . . . . . . . . . . . . . . . . . . . 343