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

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  1. Feb 1, 2013 #1

    micromass

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    Table of Contents:
    Code (Text):

    [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]
    [*] Quadratic Reciprocity
    [LIST]
    [*] Quadratic Residues
    [*] Law of Quadratic Reciprocity
    [*] A Proof of the Law of Quadratic Reciprocity
    [/LIST]
    [*] Quadratic Gauss Sums
    [LIST]
    [*] Algebraic Numbers and Algebraic Integers
    [*] The Quadratic Character of 2
    [*] Quadratic Gauss Sums
    [*] 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]
    [*] Cubic and Biquadratic Reciprocity
    [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
    [*] Biquadratic Reciprocity: Preliminaries
    [*] The Quartic Residue Symbol
    [*] The Law of Biquadratic Reciprocity
    [*] Rational 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]
    [*] Quadratic and Cyclotomic Fields
    [LIST]
    [*] Quadratic Number Fields
    [*] Cyclotomic Fields
    [*] Quadratic Reciprocity Revisited
    [/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
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