# A Classical Introduction to Modern Number Theory

Springer Science & Business Media, 17 avr. 2013 - 394 pages
Bridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves.

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### Table des matičres

 3 Unique Factorization in a Principal Ideal Domain 8 CHAPTER 17 4 The Growth of Trx 23 CHAPTER 6 26 CHAPTER 11 36 CHAPTER 4 39 2 Law of Quadratic Reciprocity 53 Quadratic Gauss Sums 66
 3 The Stickelberger Relation 207 5 The Proof of the Eisenstein Reciprocity Law 215 CHAPTER 15 223 2 Congruences Involving Bernoulli Numbers 234 3 Herbrands Theorem 241 CHAPTER 16 247 3 Dirichlet Characters 253 6 Evaluating Ls X at Negative Integers 261

 2 The Quadratic Character of 2 69 3 Quadratic Gauss Sums 70 4 The Sign of the Quadratic Gauss Sum 73 CHAPTER 7 79 2 The Existence of Finite Fields 83 3 An Application to Quadratic Residues 85 CHAPTER 8 88 2 Gauss Sums 91 3 Jacobi Sums 92 4 The Equation x + y 1 in F 97 5 More on Jacobi Sums 98 6 Applications 101 7 A General Theorem 102 CHAPTER 9 104 Cubic and Biquadratic Reciprocity 108 1 The Ring Zo 109 2 Residue Class Rings 111 3 Cubic Residue Character 12 112 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 18 118 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 10 138 2 Chevalleys Theorem 143 3 Gauss and Jacobi Sums over Finite Fields 145 The Zeta Function 151 2 Trace and Norm in Finite Fields 158 5 The Last Entry 166 CHAPTER 12 172 3 Ramification and Degree 181 Quadratic and Cyclotomic Fields 188 3 Quadratic Reciprocity Revisited 199
 Diophantine Equations 269 2 The Method of Descent 271 3 Legendres Theorem 272 4 Sophie Germains Theorem 275 5 Pells Equation 276 6 Sums of Two Squares 278 7 Sums of Four Squares 280 Exponent 3 284 9 Cubic Curves with Infinitely Many Rational Points 287 10 The Equation y2 x + k 288 11 The First Case of Fermats Conjecture for Regular Exponent 290 12 Diophantine Equations and Diophantine Approximation 292 CHAPTER 18 297 2 Local and Global Zeta Functions of an Elliptic Curve 301 3 y2 x3 + D the Local Case 304 4 y x Dx the Local Case 306 5 Hecke Lfunctions 307 6 y x Dx the Global Case 310 7 y2 x3 + D the Global Case 312 8 Final Remarks 314 CHAPTER 19 319 2 The Group E2E 323 3 The Weak Dirichlet Unit Theorem 326 4 The Weak MordellWeil Theorem 328 5 The Descent Argument 330 New Progress in Arithmetic Geometry 339 1 The Mordell Conjecture 340 2 Elliptic Curves 343 3 Modular Curves 345 4 Heights and the Height Regulator 347 5 New Results on the BirchSwinnertonDyer Conjecture 353 6 Applications to Gausss Class Number Conjecture 358 Selected Hints for the Exercises 367 Bibliography 375 Index 385 Droits d'auteur