Number Theory

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xTable of Contents2.4.1 General Properties of Finite Fields . . . . . . . . . . . . . . . . . 652.4.2 Galois Theory of Finite Fields . . . . . . . . . . . . . . . . . . . . . 692.4.3 Polynomials over Finite Fields . . . . . . . . . . . . . . . . . . . . . 712.5 Bounds for the Number of Solutions in Finite Fields . . . . . . . . 722.5.1 The Chevalley–Warning Theorem . . . . . . . . . . . . . . . . . . 722.5.2 Gauss Sums for Finite Fields. . . . . . . . . . . . . . . . . . . . . . . 742.5.3 Jacobi Sums for Finite Fields . . . . . . . . . . . . . . . . . . . . . . 792.5.4 The Jacobi Sums J(χ 1 , χ 2 ) . . . . . . . . . . . . . . . . . . . . . . . . 832.5.5 The Number of Solutions of Diagonal Equations . . . . . . 872.5.6 The Weil Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.5.7 The Weil Conjectures (Deligne’s Theorem) . . . . . . . . . . 922.6 Exercises for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933. Basic Algebraic Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1 Field-Theoretic Algebraic Number Theory . . . . . . . . . . . . . . . . . 1013.1.1 Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.1.2 Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.1.4 Characteristic Polynomial, Norm, Trace . . . . . . . . . . . . . 1093.1.5 Noether’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.1.6 The Basic Theorem of Kummer Theory . . . . . . . . . . . . . 1113.1.7 Examples of the Use of Kummer Theory . . . . . . . . . . . . 1143.1.8 Artin–Schreier Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.2 The Normal Basis Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173.2.1 Linear Independence and Hilbert 90 . . . . . . . . . . . . . . . . 1173.2.2 The Normal Basis Theorem in the Cyclic Case . . . . . . . 1193.2.3 Additive Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203.2.4 Algebraic Independence of Homomorphisms . . . . . . . . . 1213.2.5 The Normal Basis Theorem. . . . . . . . . . . . . . . . . . . . . . . . 1233.3 Ring-Theoretic Algebraic Number Theory . . . . . . . . . . . . . . . . . 1243.3.1 Gauss’s Lemma on Polynomials . . . . . . . . . . . . . . . . . . . . 1243.3.2 Algebraic Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.3.3 Ring of Integers and Discriminant . . . . . . . . . . . . . . . . . . 1283.3.4 Ideals and Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1303.3.5 Decomposition of Primes and Ramification . . . . . . . . . . 1323.3.6 Galois Properties of Prime Decomposition . . . . . . . . . . . 1343.4 Quadratic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1363.4.1 Field-Theoretic and Basic Ring-Theoretic Properties . . 1363.4.2 Results and Conjectures on Class and Unit Groups . . . 1383.5 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.5.1 Cyclotomic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.5.2 Field-Theoretic Properties of Q(ζ n ) . . . . . . . . . . . . . . . . . 1443.5.3 Ring-Theoretic Properties . . . . . . . . . . . . . . . . . . . . . . . . . 1463.5.4 The Totally Real Subfield of Q(ζ p k). . . . . . . . . . . . . . . . . 1483.6 Stickelberger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
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 Number 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 Number 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
Page 5 and 6: vKoblitz formula for Morita’s p-a
Page 8: viiiinstance) A(X) = A d X d +A d
Page 13 and 14: Table of Contentsxiii6.3.1 The Gene
Page 15 and 16: Table of Contentsxv8.2.3 The Fundam
Page 17 and 18: Table of Contentsxvii10.2.2 The Use
Page 19 and 20: Table of Contentsxix12.8.3 A Genera
Page 21: Table of Contentsxxi16.4 Mihăilesc

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