# Number Theory A Classical Introduction to Modern Number Theory by Ireland and Rosen

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1. Feb 1, 2013

### micromass

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[LIST]
[*] Preface
[*] Unique Factorization
[LIST]
[*] Unique Factorization in Z
[*] Unique Factorization in k[x]
[*] Unique Factorization in a Principal Ideal Domain
[*] The Rings Z[i] and Z[\omega]
[/LIST]
[*] Applications of Unique Factorization
[LIST]
[*] Infinitely Many Primes in Z
[*] Some Arithmetic Functions
[*] \sum 1/p Diverges
[*] The Growth of \pi(x)
[/LIST]
[*] Congruence
[LIST]
[*] Elementary Observations
[*] Congruence in Z
[*] The Congruence ax = b(m)
[*] The Chinese Remainder Theorem
[/LIST]
[*] The Structure of U(Z/nZ)
[LIST]
[*] Primitive Roots and the Group Structure of U(Z/nZ)
[*] nth Power Residues
[/LIST]
[LIST]
[*] A Proof of the Law of Quadratic Reciprocity
[/LIST]
[LIST]
[*] Algebraic Numbers and Algebraic Integers
[*] The Quadratic Character of 2
[*] The Sign of the Quadratic Gauss Sum
[/LIST]
[*] Finite Fields
[LIST]
[*] Basic Properties of Finite Fields
[*] The Existence of Finite Fields
[*] An Application to Quadratic Residues
[/LIST]
[*] Gauss and Jacobi Sums
[LIST]
[*] Multiplicative Characters
[*] Gauss Sums
[*] Jacobi Sums
[*] The Equation x^n + y^n = 1 in F_p
[*] More on Jacobi Sums
[*] Applications
[*] A General Theorem
[/LIST]
[LIST]
[*] The Ring Z[\omega]
[*] 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 2
[*] The Quartic Residue Symbol
[*] The Law of Biquadratic Reciprocity
[*] The Constructibility of Regular Polygons
[*] Cubic Gauss Sums and the Problem of Kummer
[/LIST]
[*] Equations over Finite Fields
[LIST]
[*] Affine Space, Projective Space, and Polynomials
[*] Chevalley's Theorem
[*] Gauss and Jacobi Sums over Finite Fields
[/LIST]
[*] The Zeta Function
[LIST]
[*] The Zeta Function of a Projective Hypersurface
[*] Trace and Norm in Finite Fields
[*] The Rationality of the Zeta Function Associated to a_0x_0^m + a_1x_1^m + ... + a_nx_n^m
[*] A Proof of the Hasse-Davenport Relation
[*] The Last Entry
[/LIST]
[*] Algebraic Number Theory
[LIST]
[*] Algebraic Preliminaries
[*] Unique Factorization in Algebraic Number Fields
[*] Ramification and Degree
[/LIST]
[LIST]
[*] Cyclotomic Fields
[/LIST]
[*] The Stickelberger Relation and the Eisenstein Reciprocity Law
[LIST]
[*] 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
[/LIST]
[*] Bernoulli Numbers
[LIST]
[*] Bernoulli Numbers; Definitions and Applications
[*] Congruences Involving Bernoulli Numbers
[*] Herbrand's Theorem
[/LIST]
[*] Dirichlet L-functions
[LIST]
[*] The Zeta Function
[*] A Special Case
[*] Dirichlet Characters
[*] Dirichlet L-functions
[*] The Key Step
[*] Evaluating L(s,\chi) at Negative Integers
[/LIST]
[*] Diophantine Equations
[LIST]
[*] 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 3
[*] Cubic Curves with Infinitely Many Rational Points
[*] The Equation y^2 = x^3 + k
[*] The First Case of Fermat's Conjecture for Regular Exponent
[*] Diophantine Equations and Diophantine Approximation
[/LIST]
[*] Elliptic Curves
[LIST]
[*] 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
[/LIST]
[*] The Mordell-Weil Theorem
[LIST]
[*] The Addition Law and Several Identities
[*] The Group E/2E
[*] The Weak Dirichlet Unit Theorem
[*] The Weak Mordell-Weil Theorem
[*] The Descent Argument
[/LIST]
[*] New Progress in Arithmetic Geometry
[LIST]
[*] 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
[/LIST]
[*] Selected Hints for the Exercises
[*] Bibliography
[*] Index
[/LIST]

Last edited: Feb 3, 2013