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［ω］ 12
CHAPTER 2 Applications of Unique Factorization 17
1 Infinitely Many Primes in Z 17
2 Some Arithmetic Functions 18
3 ∑ 1/p Diverges 21
4 The Growth of π（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 U（Z/nZ） 39
2 nth Power Residues 45
CHAPTER 5 Quadratic Reciprocity 50
1 Quadratic Residues 50
2 Law of Quadratic Reciprocity 53
3 A Proof of the Law of Quadratic Reciprocity 58
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+yn=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［ω］ 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
CHAPTER 11 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 a0xm 0+a1xm 1+…+anxm n 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 Bernouilli 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,x）at Negative Integers 261
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+k 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=x3+D,the Local Case 304
4 y2=x3-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 E/2E 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 345
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