.

0 (hodnocen0 x )

(1) Půjčeno:2x 

BK

2nd ed.

New York : Springer, 1990

ivx,389 s.

ISBN 0-387-97329-X (váz.)

Obsahuje rejstřík

Bibliografie na s. 375-384

Čísla - teorie čísel - učebnice vysokošk.

000014059

Contents // Preface to the Second Edition v // Preface vii // Chapter 1 // Unique Factorization 1 // §1 Unique Factorization in Z 1 // §2 Unique Factorization in k[x) 6 // §3 Unique Factorization in a Principal Ideal Domain 8 // §4 The Rings Z[i] and Z[a>] 12 // Chapter 2 // Applications of Unique Factorization 17 // §1 Infinitely Many Primes in Z 17 // §2 Some Arithmetic Functions 18 // §3 2 lip Diverges 21 // §4 The Growth of ir(x) 22 // Chapter 3 // Congruence 28 // §l Elementary Observations 28 // §2 Congruence in Z 29 // §3 The Congruence ax = b(m) 31 // §4 The Chinese Remainder Theorem 34 // Chapter 4 // The Structure of L/(Z/nZ) 39 // §l Primitive Roots and the Group Structure of U(llnl) 39 // §2 nth Power Residues 45 // Chapter 5 // Quadratic Reciprocity 50 // §l Quadratic Residues 50 // §2 Law of Quadratic Reciprocity 53 // §3 A Proof of the Law of Quadratic Reciprocity 58 // xi // xii // Contents // Chapter 6 // Quadratic Gauss Sums 66 // §1 Algebraic Numbers and Algebraic Integers 66 // §2 The Quadratic Character of 2 69 // §3 Quadratic Gauss Sums 70 // §4 The Sign of the Quadratic Gauss Sum 73 // Chapter 7 // Finite Fields 79 // §1 Basic Properties of Finite Fields 79 // §2 The Existence of Finite Fields 83 // §3 An Application to Quadratic Residues 85 // Chapter 8 // Gauss and Jacobi Sums 88 // § 1 Multiplicative Characters 88 // §2 Gauss Sums 91 // §3 Jacobi Sums 92 // §4 The Equation xn + y" = 1 in Fp 97 // §5 More on Jacobi Sums 98

// §6 Applications 101 // §7 A General Theorem 102 // Chapter 9 // Cubic and Biquadratic Reciprocity 108 // §1 The Ring Z[oj] 109 // §2 Residue Class Rings 111 // §3 Cubic Residue Character 112 // §4 Proof of the Law of Cubic Reciprocity 115 // §5 Another Proof of the Law of Cubic Reciprocity 117 // §6 The Cubic Character of 2 118 // §7 Biquadratic Reciprocity: Preliminaries 119 // §8 The Quartic Residue Symbol 121 // §9 The Law of Biquadratic Reciprocity 123 // §10 Rational Biquadratic Reciprocity 127 // §11 The Constructibility of Regular Polygons 130 // §12 Cubic Gauss Sums and the Problem of Kummer 131 // Chapter 10 // Equations over Finite Fields 138 // §1 Affine Space, Projective Space, and Polynomials 138 // §2 Chevalley’s Theorem 143 // §3 Gauss and Jacobi Sums over Finite Fields 145 // Contents // xiii // Chapter II // The Zeta Function 151 // §1 The Zeta Function of a Projective Hypersurface 151 // §2 Trace and Norm in Finite Fields 158 // §3 The Rationality of the Zeta Function Associated to // aoxo + atx\\ + • ■ ■ + anx™ 161 // §4 A Proof of the Hasse-Davenport Relation 163 // §5 The Last Entry 166 // Chapter 12 // Algebraic Number Theory 172 // §1 Algebraic Preliminaries 172 // §2 Unique Factorization in Algebraic Number Fields 174 // §3 Ramification and Degree 181 // Chapter 13 // Quadratic and Cyclotomic Fields 188 // §1 Quadratic Number Fields 188 // §2 Cyclotomic Fields 193 // §3 Quadratic Reciprocity Revisited 199 // Chapter

14 // The Stickelberger Relation and the Eisenstein Reciprocity Law 203 // §1 The Norm of an Ideal 203 // §2 The Power Residue Symbol 204 // §3 The Stickelberger Relation 207 // §4 The Proof of the Stickelberger Relation 209 // §5 The Proof of the Eisenstein Reciprocity Law 215 // §6 Three Applications 220 // Chapter 15 // Bernoulli Numbers 228 // §1 Bernoulli Numbers; Definitions and Applications 228 // §2 Congruences Involving Bernoulli Numbers 234 // §3 Herbrand’s Theorem 241 // Chapter 16 // Dirichlet L-functions 249 // §1 The Zeta Function 249 // §2 A Special Case 251 // §3 Dirichlet Characters 253 // §4 Dirichlet L-functions 255 // §5 The Key Step 257 // §6 Evaluating L(s, \\) at Negative Integers 261 // xiv // Contents // Chapter 17 // Diophantine Equations 269 // §1 Generalities and First Examples 269 // §2 The Method of Descent 271 // §3 Legendre’s Theorem 272 // §4 Sophie Germain’s Theorem 275 // §5 Pell’s Equation 276 // §6 Sums of Two Squares 278 // §7 Sums of Four Squares 280 // §8 The Fermat Equation: Exponent 3 284 // §9 Cubic Curves with Infinitely Many Rational Points 287 // §10 The Equation y2 = x3 + к 288 // § 11 The First Case of Fermat’s Conjecture for Regular Exponent 290 // §12 Diophantine Equations and Diophantine Approximation 292 // Chapter 18 // Elliptic Curves 297 // §1 Generalities 297 // §2 Local and Global Zeta Functions of an Elliptic Curve 301 // §3 y2 = Xі + D, the Local Case 304 // §4 у2 = хъ - Dx,

the Local Case 306 // §5 Hecke L-functions 307 // §6 y2 = x3 — Dx, the Global Case 310 // §7 y2 = x3 + D, the Global Case 312 // §8 Final Remarks 314 // Chapter 19 // The Mordell-Weil Theorem 319 // §1 The Addition Law and Several Identities 320 // §2 The Group EHE 323 // §3 The Weak Dirichlet Unit Theorem 326 // §4 The Weak Mordell-Weil Theorem 328 // §5 The Descent Argument 330 // Chapter 20 // New Progress in Arithmetic Geometry 339 // §1 The Mordell Conjecture 340 // §2 Elliptic Curves 343 // §3 Modular Curves 343 // §4 Heights and the Height Regulator 348 // §5 New Results on the Birch-Swinnerton-Dyer Conjecture 353 // §6 Applications to Gauss’s Class Number Conjecture 358 // Selected Hints for the Exercises 367 // Bibliography 375 // Index 385