目錄
Preface to the Second Edition
Preface
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[w]
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[w]
§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 Afine 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 aoxm0+anxm1+…+anxmn
§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 Bernoulli 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, χ)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
[