现代数论经典引论处 第2版【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

现代数论经典引论处 第2版PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

CHAPTER 1 Unique Factorization1

1 Unique Factorization in Z1

2 Unique Factorization in k［x］6

3 Unique Factorization in a Principal Ideal Domain8

4 The Rings Z［i］and Z［ω］12

CHAPTER 2 Applications of Unique Factorization17

1 Infinitely Many Primes in Z17

2 Some Arithmetic Functions18

3 ∑ 1/p Diverges21

4 The Growth of π（x）22

CHAPTER 3 Congruence28

1 Elementary Observations28

2 Congruence in Z29

3 The Congruence ax？b（m）31

4 The Chinese Remainder Theorem34

CHAPTER 4 The Structure of U（Z/nZ）39

1 Primitive Roots and the Group Structure of U（Z/nZ）39

2 nth Power Residues45

CHAPTER 5 Quadratic Reciprocity50

1 Quadratic Residues50

2 Law of Quadratic Reciprocity53

3 A Proof of the Law of Quadratic Reciprocity58

CHAPTER 6 Quadratic Gauss Sums66

1 Algebraic Numbers and Algebraic Integers66

2 The Quadratic Character of 269

3 Quadratic Gauss Sums70

4 The Sign of the Quadratic Gauss Sum73

CHAPTER 7 Finite Fields79

1 Basic Properties of Finite Fields79

2 The Existence of Finite Fields83

3 An Application to Quadratic Residues85

CHAPTER 8 Gauss and Jacobi Sums88

1 Multiplicative Characters88

2 Gauss Sums91

3 Jacobi Sums92

4 The Equation xn+yn=1 in Fp97

5 More on Jacobi Sums98

6 Applications101

7 A General Theorem102

CHAPTER 9 Cubic and Biquadratic Reciprocity108

1 The Ring Z［ω］109

2 Residue Class Rings111

3 Cubic Residue Character112

4 Proof of the Law of Cubic Reciprocity115

5 Another Proof of the Law of Cubic Reciprocity117

6 The Cubic Character of 2118

7 Biquadratic Reciprocity：Preliminaries119

8 The Quartic Residue Symbol121

9 The Law of Biquadratic Reciprocity123

10 Rational Biquadratic Reciprocity127

11 The Constructibility of Regular Polygons130

12 Cubic Gauss Sums and the Problem of Kummer131

CHAPTER 10 Equations over Finite Fields138

1 Affine Space,Projective Space,and Polynomials138

2 Chevalley's Theorem143

3 Gauss and Jacobi Sums over Finite Fields145

CHAPTER 11 The Zeta Function151

1 The Zeta Function of a Projective Hypersurface151

2 Trace and Norm in Finite Fields158

3 The Rationality of the Zeta Function Associated to a0xm 0+a1xm 1+…+anxm n161

4 A Proof of the Hasse-Davenport Relation163

5 The Last Entry166

CHAPTER 12 Algebraic Number Theory172

1 Algebraic Preliminaries172

2 Unique Factorization in Algebraic Number Fields174

3 Ramification and Degree181

CHAPTER 13 Quadratic and Cyclotomic Fields188

1 Quadratic Number Fields188

2 Cyclotomic Fields193

3 Quadratic Reciprocity Revisited199

CHAPTER 14 The Stickelberger Relation and the Eisenstein Reciprocity Law203

1 The Norm of an Ideal203

2 The Power Residue Symbol204

3 The Stickelberger Relation207

4 The Proof of the Stickelberger Relation209

5 The Proof of the Eisenstein Reciprocity Law215

6 Three Applications220

CHAPTER 15 Bernouilli Numbers228

1 Bernoulli Numbers；Definitions and Applications228

2 Congruences Involving Bernoulli Numbers234

3 Herbrand's Theorem241

CHAPTER 16 Dirichlet L-functions249

1 The Zeta Function249

2 A Special Case251

3 Dirichlet Characters253

4 Dirichlet L-functions255

5 The Key Step257

6 Evaluating L（s,x）at Negative Integers261

CHAPTER 17 Diophantine Equations269

1 Generalities and First Examples269

2 The Method of Descent271

3 Legendre's Theorem272

4 Sophie Germain's Theorem275

5 Pell's Equation276

6 Sums of Two Squares278

7 Sums of Four Squares280

8 The Fermat Equation：Exponent 3284

9 Cubic Curves with Infinitely Many Rational Points287

10 The Equation y2=x3+k288

11 The First Case of Fermat's Conjecture for Regular Exponent290

12 Diophantine Equations and Diophantine Approximation292

CHAPTER 18 Elliptic Curves297

1 Generalities297

2 Local and Global Zeta Functions of an Elliptic Curve301

3 y2=x3+D,the Local Case304

4 y2=x3-Dx,the Local Case306

5 Hecke L-functions307

6 y2=x3-Dx,the Global Case310

7 y2=x3+D,the Global Case312

8 Final Remarks314

CHAPTER 19 The Mordell-Weil Theorem319

1 The Addition Law and Several Identities320

2 The Group E/2E323

3 The Weak Dirichlet Unit Theorem326

4 The Weak Mordell-Weil Theorem328

5 The Descent Argument330

CHAPTER 20 New Progress in Arithmetic Geometry339

1 The Mordell Conjecture340

2 Elliptic Curves343

3 Modular Curves345

4 Heights and the Height Regulator348

5 New Results on the Birch-Swinnerton-Dyer Conjecture353

6 Applications to Gauss's Class Number Conjecture358

Selected Hints for the Exercises367

Bibliography375

Index385