A Classical Introduction to Modern Number Theory

Transcription

1 Kenneth Ireland Michael Rosen A Classical Introduction to Modern Number Theory Second Edition Springer

2 Contents Preface to the Second Edition Preface v 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[w] 12 CHAPTER 2 Applications of Unique Factorization 17 1 Infinitely Many Primes in Z 17 2 Some Arithmetic Functions \lp Diverges 21 4 The Growth of ir(x) 22 CHAPTER 3 Congruence 28 1 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 U(Z/nZ) 39 1 Primitive Roots and the Group Structure of C/(Z/nZ) 39 2 «th Power Residues 45 CHAPTER 5 Quadratic Reciprocity Quadratic Residues 50 2 Law of Quadratic Reciprocity 53 3 A Proof of the Law of Quadratic Reciprocity 58 XI

3 Xll Contents CHAPTER 6 Quadratic Gauss Sums 66 1 Algebraic Numbers and Algebraic Integers 66 2 The Quadratic Character of 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 x" + y" = 1 in F p 97 5 More on Jacobi Sums 98 6 Applications A General Theorem 102 CHAPTER 9 Cubic and Biquadratic Reciprocity The Ring Z[o>] Residue Class Rings Cubic Residue Character Proof of the Law of Cubic Reciprocity Another Proof of the Law of Cubic Reciprocity The Cubic Character of Biquadratic Reciprocity: Preliminaries The Quartic Residue Symbol The Law of Biquadratic Reciprocity Rational Biquadratic Reciprocity The Constructibility of Regulär Polygons Cubic Gauss Sums and the Problem of Kummer 131 CHAPTER 10 Equations over Finite Fields Affine Space, Projective Space, and Polynomials Chevalley's Theorem Gauss and Jacobi Sums over Finite Fields 145

4 Contents xiii CHAPTER 11 The Zeta Function The Zeta Function of a Projective Hypersurface Trace and Norm in Finite Fields The Rationality of the Zeta Function Associated to flojcs 1 + aixt + + a n x A Proof of the Hasse-Davenport Relation The Last Entry 166 CHAPTER 12 Algebraic Number Theory Algebraic Preliminaries Unique Factonzation in Algebraic Number Fields Ramification and Degree 181 CHAPTER 13 Quadratic and Cyclotomic Fields Quadratic Number Fields Cyclotomic Fields Quadratic Reciprocity Revisited 199 CHAPTER 14 The Stickelberger Relation and the Eisenstein Reciprocity Law The Norm of an Ideal The Power Residue Symbol The Stickelberger Relation The Proof of the Stickelberger Relation The Proof of the Eisenstein Reciprocity Law Three Applications 220 CHAPTER 15 Bernoulli Numbers Bernoulli Numbers; Definitions and Applications Congruences Involving Bernoulli Numbers Herbrand's Theorem 241 CHAPTER 16 Dirichlet L-functions The Zeta Function A Special Case Dirichlet Characters Dirichlet L-functions The Key Step Evaluating L(s, x) at Negative Integers 261

5 XIV Contents CHAPTER 17 Diophantine Equations Generalities and First Examples The Method of Descent ' Legendre's Theorem Sophie Germain's Theorem Pell's Equation Sums of Two Squares Sums of Four Squares The Fermat Equation: Exponent Cubic Curves with Infinitely Many Rational Points The Equation y 2 = x 3 + k The First Case of Fermat's Conjecture for Regulär Exponent Diophantine Equations and Diophantine Approximation 292 CHAPTER 18 Elliptic Curves Generalities Local and Global Zeta Functions of an Elliptic Curve y 2 = x 3 + D, the Local Case y 2 = x 3 - Dx, the Local Case Hecke L-functions y 2 = x 3 - Dx, the Global Case y 2 = x 3 + D, the Global Case Final Remarks 314 CHAPTER 19 The Mordeil-Weil Theorem The Addition Law and Several Identities The Group EHE The Weak Dirichlet Unit Theorem The Weak Mordell-Weil Theorem The Descent Argument 330 CHAPTER 20 New Progress in Arithmetic Geometry The Mordell Conjecture Elliptic Curves Modular Curves Heights and the Height Regulator New Results on the Birch-Swinnerton-Dyer Conjecture Applications to Gauss's Class Number Conjecture 358 Selected Hints for the Exercises 367 Bibliography 375 Index 385