Number Theory
Number Theory
Number Theory
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Table of Contentsxi3.6.1 Introduction and Algebraic Setting . . . . . . . . . . . . . . . . . 1503.6.2 Instantiation of Gauss Sums . . . . . . . . . . . . . . . . . . . . . . . 1513.6.3 Prime Ideal Decomposition of Gauss Sums. . . . . . . . . . . 1543.6.4 The Stickelberger Ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.6.5 Diagonalization of the Stickelberger Element . . . . . . . . . 1633.6.6 The Eisenstein Reciprocity Law . . . . . . . . . . . . . . . . . . . . 1653.7 The Hasse–Davenport Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.7.1 Distribution Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1713.7.2 The Hasse–Davenport Relations . . . . . . . . . . . . . . . . . . . . 1733.7.3 The Zeta Function of a Diagonal Hypersurface . . . . . . . 1773.8 Exercises for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1794. p-adic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.1 Absolute Values and Completions . . . . . . . . . . . . . . . . . . . . . . . . 1854.1.1 Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.1.2 Archimedean Absolute Values . . . . . . . . . . . . . . . . . . . . . . 1864.1.3 Non-Archimedean and Ultrametric Absolute Values . . . 1904.1.4 Ostrowski’s Theorem and the Product Formula . . . . . . 1924.1.5 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1944.1.6 Completions of a <strong>Number</strong> Field . . . . . . . . . . . . . . . . . . . . 1974.1.7 Hensel’s Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2014.2 Analytic Functions in p-adic Fields . . . . . . . . . . . . . . . . . . . . . . . 2074.2.1 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.2.2 Examples of Analytic Functions . . . . . . . . . . . . . . . . . . . . 2104.2.3 Application of the Artin–Hasse Exponential . . . . . . . . . 2194.2.4 Mahler Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2224.3 Additive and Multiplicative Structures . . . . . . . . . . . . . . . . . . . . 2264.3.1 Concrete Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2264.3.2 Basic Reductions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2284.3.3 Study of the Groups U i . . . . . . . . . . . . . . . . . . . . . . . . . . . 2314.3.4 Study of the Group U 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2344.3.5 The Group K ∗ p/K ∗ p 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364.4 Extensions of p-adic Fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2374.4.1 Preliminaries on Local Field Norms . . . . . . . . . . . . . . . . . 2384.4.2 Krasner’s Lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2414.4.3 General Results on Extensions . . . . . . . . . . . . . . . . . . . . . 2424.4.4 Applications of the Cohomology of Cyclic Groups . . . . 2454.4.5 Characterization of Unramified Extensions. . . . . . . . . . . 2514.4.6 Properties of Unramified Extensions . . . . . . . . . . . . . . . . 2534.4.7 Totally Ramified Extensions . . . . . . . . . . . . . . . . . . . . . . . 2554.4.8 Analytic Representations of pth Roots of Unity . . . . . . 2574.4.9 Factorizations in <strong>Number</strong> Fields . . . . . . . . . . . . . . . . . . . . 2604.4.10 Existence of the Field C p . . . . . . . . . . . . . . . . . . . . . . . . . . 2624.4.11 Some Analysis in C p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2654.5 The Theorems of Strassmann and Weierstrass . . . . . . . . . . . . . . 268