CHAPTER 1 Unique Factorization

1 Unique Factorization in Z

2 Unique Factorization in k［x］

3 Unique Factorization in a Principal Ideal Domain

4 The Rings Z［i］and Z［ω］

CHAPTER 2 Applications of Unique Factorization

1 Infinitely Many Primes in Z

2 Some Arithmetic Functions

3 ∑ 1/p Diverges

4 The Growth of π（x）

CHAPTER 3 Congruence

1 Elementary Observations

2 Congruence in Z

3 The Congruence ax？b（m）

4 The Chinese Remainder Theorem

CHAPTER 4 The Structure of U（Z/nZ）

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

2 nth Power Residues

CHAPTER 5 Quadratic Reciprocity

1 Quadratic Residues

2 Law of Quadratic Reciprocity

3 A Proof of the Law of Quadratic Reciprocity

CHAPTER 6 Quadratic Gauss Sums

1 Algebraic Numbers and Algebraic Integers

2 The Quadratic Character of 2

3 Quadratic Gauss Sums

4 The Sign of the Quadratic Gauss Sum

CHAPTER 7 Finite Fields

1 Basic Properties of Finite Fields

2 The Existence of Finite Fields

3 An Application to Quadratic Residues

CHAPTER 8 Gauss and Jacobi Sums

1 Multiplicative Characters

2 Gauss Sums

3 Jacobi Sums

4 The Equation xn+yn=1 in Fp

5 More on Jacobi Sums

6 Applications

7 A General Theorem

CHAPTER 9 Cubic and Biquadratic Reciprocity

1 The Ring Z［ω］

2 Residue Class Rings

3 Cubic Residue Character

4 Proof of the Law of Cubic Reciprocity

5 Another Proof of the Law of Cubic Reciprocity

6 The Cubic Character of 2

7 Biquadratic Reciprocity：Preliminaries

8 The Quartic Residue Symbol

9 The Law of Biquadratic Reciprocity

10 Rational Biquadratic Reciprocity

11 The Constructibility of Regular Polygons

12 Cubic Gauss Sums and the Problem of Kummer

CHAPTER 10 Equations over Finite Fields

1 Affine Space,Projective Space,and Polynomials

2 Chevalley's Theorem

3 Gauss and Jacobi Sums over Finite Fields

CHAPTER 11 The Zeta Function

1 The Zeta Function of a Projective Hypersurface

2 Trace and Norm in Finite Fields

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

4 A Proof of the Hasse-Davenport Relation

5 The Last Entry

CHAPTER 12 Algebraic Number Theory

1 Algebraic Preliminaries

2 Unique Factorization in Algebraic Number Fields

3 Ramification and Degree

CHAPTER 13 Quadratic and Cyclotomic Fields

1 Quadratic Number Fields

2 Cyclotomic Fields

3 Quadratic Reciprocity Revisited

CHAPTER 14 The Stickelberger Relation and the Eisenstein Reciprocity Law

1 The Norm of an Ideal

2 The Power Residue Symbol

3 The Stickelberger Relation

4 The Proof of the Stickelberger Relation

5 The Proof of the Eisenstein Reciprocity Law

6 Three Applications

CHAPTER 15 Bernouilli Numbers

1 Bernoulli Numbers；Definitions and Applications

2 Congruences Involving Bernoulli Numbers

3 Herbrand's Theorem

CHAPTER 16 Dirichlet L-functions

1 The Zeta Function

2 A Special Case

3 Dirichlet Characters

4 Dirichlet L-functions

5 The Key Step

6 Evaluating L（s,x）at Negative Integers

CHAPTER 17 Diophantine Equations

1 Generalities and First Examples

2 The Method of Descent

3 Legendre's Theorem

4 Sophie Germain's Theorem

5 Pell's Equation

6 Sums of Two Squares

7 Sums of Four Squares

8 The Fermat Equation：Exponent 3

9 Cubic Curves with Infinitely Many Rational Points

10 The Equation y2=x3+k

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

12 Diophantine Equations and Diophantine Approximation

CHAPTER 18 Elliptic Curves

1 Generalities

2 Local and Global Zeta Functions of an Elliptic Curve

3 y2=x3+D,the Local Case

4 y2=x3-Dx,the Local Case

5 Hecke L-functions

6 y2=x3-Dx,the Global Case

7 y2=x3+D,the Global Case

8 Final Remarks

CHAPTER 19 The Mordell-Weil Theorem

1 The Addition Law and Several Identities

2 The Group E/2E

3 The Weak Dirichlet Unit Theorem

4 The Weak Mordell-Weil Theorem

5 The Descent Argument

CHAPTER 20 New Progress in Arithmetic Geometry

1 The Mordell Conjecture

2 Elliptic Curves

3 Modular Curves

4 Heights and the Height Regulator

5 New Results on the Birch-Swinnerton-Dyer Conjecture

6 Applications to Gauss's Class Number Conjecture

Selected Hints for the Exercises

Bibliography

Index