Primes of the Form X2+ny2 : Fermat, Class Field Theory, and Complex Multiplication

[BOOK DESCRIPTION]

An exciting approach to the history and mathematics of number theory "...the author's style is totally lucid and very easy to read ...the result is indeed a wonderful story." -Mathematical Reviews Written in a unique and accessible style for readers of varied mathematical backgrounds, the Second Edition of Primes of the Form p = x2+ ny2 details the history behind how Pierre de Fermat's work ultimately gave birth to quadratic reciprocity and the genus theory of quadratic forms. The book also illustrates how results of Euler and Gauss can be fully understood only in the context of class field theory, and in addition, explores a selection of the magnificent formulas of complex multiplication. Primes of the Form p = x2 + ny2, Second Edition focuses on addressing the question of when a prime p is of the form x2 + ny2, which serves as the basis for further discussion of various mathematical topics. This updated edition has several new notable features, including: * A well-motivated introduction to the classical formulation of class field theory * Illustrations of explicit numerical examples to demonstrate the power of basic theorems in various situations * An elementary treatment of quadratic forms and genus theory * Simultaneous treatment of elementary and advanced aspects of number theory * New coverage of the Shimura reciprocity law and a selection of recent work in an updated bibliography Primes of the Form p = x2 + ny2, Second Edition is both a useful reference for number theory theorists and an excellent text for undergraduate and graduate-level courses in number and Galois theory.

[TABLE OF CONTENTS]

Preface To The First Edition                       ix
Preface To The Second Edition                      xi
Notation                                           xiii
Introduction                                       1   (6)
Chapter One From Fermat To Gauss
  ｧ1 Fermat, Euler And Quadratic Reciprocity       7   (15)
    A Fermat                                       8   (1)
    B Euler                                        9   (2)
    C p = xｲ + nyｲ and Quadratic Reciprocity       11  (6)
    D Beyond Quadratic Reciprocity                 17  (2)
    E Exercises                                    19  (3)
  ｧ2 Lagrange, Legendre And Quadratic Forms        22  (20)
    A Quadratic Forms                              22  (5)
    B p = xｲ + nyｲ and Quadratic Forms             27  (3)
    C Elementary Genus Theory                      30  (4)
    D Lagrange and Legendre                        34  (5)
    E Exercises                                    39  (3)
  ｧ3 Gauss, Composition And Genera                 42  (25)
    A Composition and the Class Group              43  (5)
    B Genus Theory                                 48  (5)
    C p = xｲ + nyｲ and Euler's Convenient          53  (4)
    Numbers
    D Disquisitiones Arithmeticae                  57  (2)
    E Exercises                                    59  (8)
  ｧ4 Cubic And Biquadratic Reciprocity             67  (20)
    A Z[ω] and Cubic Reciprocity             67  (6)
    B Z[i] and Biquadratic Reciprocity             73  (2)
    C Gauss and Higher Reciprocity                 75  (5)
    D Exercises                                    80  (7)
Chapter Two Class Field Theory
  ｧ5 The Hilbert Class Field And p = xｲ + nyｲ      87  (21)
    A Number Fields                                88  (4)
    B Quadratic Fields                             92  (2)
    C The Hilbert Class Field                      94  (4)
    D Solution of p = xｲ + nyｲ for Infinitely      98  (5)
    Many n
    E Exercises                                    103 (5)
  ｧ6 The Hilbert Class Field And Genus Theory      108 (12)
    A Genus Theory for Field Discriminants         109 (5)
    B Applications to the Hilbert Class Field      114 (2)
    C Exercises                                    116 (4)
  ｧ7 Orders In Imaginary Quadratic Fields          120 (24)
    A Orders in Quadratic Fields                   120 (3)
    B Orders and Quadratic Forms                   123 (6)
    C Ideals Prime to the Conductor                129 (3)
    D The Class Number                             132 (4)
    E Exercises                                    136 (8)
  ｧ8 Class Field Theory And The Cebotarev          144 (18)
  Density Theorem
    A The Theorems of Class Field Theory           144 (8)
    B The Cebotarev Density Theorem                152 (4)
    C Norms and Ideles                             156 (1)
    D Exercises                                    157 (5)
  ｧ9 Ring Class Fields And p = xｲ + nyｲ            162 (19)
    A Solution of p = xｲ + nyｲ for All n           162 (4)
    B The Ring Class Fields of Z[Square root of    166 (4)
    -27] and Z[Square root of -64]
    C Primes Represented by Positive Definite      170 (2)
    Quadratic Forms
    D Ring Class Fields and Generalized            172 (2)
    Dihedral Extensions
    E Exercises                                    174 (7)
Chapter Three Complex Multiplication
  ｧ10 Elliptic Functions And Complex               181 (19)
  Multiplication
    A Elliptic Functions and the Weierstrass       182 (5)
    p-Function
    B The j-Invariant of a Lattice                 187 (3)
    C Complex Multiplication                       190 (7)
    D Exercises                                    197 (3)
  ｧ11 Modular Functions And Ring Class Fields      200 (26)
    A The j-Function                               200 (5)
    B Modular Functions for Γ0(m)            205 (5)
    C The Modular Equation Φm(X,Y)             210 (4)
    D Complex Multiplication and Ring Class        214 (6)
    Fields
    E Exercises                                    220 (6)
  ｧ12 Modular Functions And Singular               226 (35)
  j-Invariants
    A The Cube Root of the j-Function              226 (6)
    B The Weber Functions                          232 (5)
    C j-Invariants of Orders of Class Number 1     237 (2)
    D Weber's Computation of j(Square root of      239 (8)
    -14)
    E Imaginary Quadratic Fields of Class          247 (3)
    Number 1
    F Exercises                                    250 (11)
  ｧ13 The Class Equation                           261 (22)
    A Computing the Class Equation                 262 (6)
    B Computing the Modular Equation               268 (4)
    C Theorems of Deuring, Gross and Zagier        272 (5)
    D Exercises                                    277 (6)
Chapter Four Additional Topics
  ｧ14 Elliptic Curves                              283 (26)
    A Elliptic Curves and Weierstrass Equations    284 (3)
    B Complex Multiplication and Elliptic Curves   287 (3)
    C Elliptic Curves over Finite Fields           290 (7)
    D Elliptic Curve Primality Tests               297 (7)
    E Exercises                                    304 (5)
  ｧ15 Shimura Reciprocity                          309 (34)
    A Modular Functions and Shimura Reciprocity    309 (4)
    B Extended Ring Class Fields                   313 (2)
    C Shimura Reciprocity for Extended Ring        315 (3)
    Class Fields
    D Shimura Reciprocity for Ring Class Fields    318 (6)
    E The Idelic Approach                          324 (4)
    F Exercises                                    328 (7)
References                                         335 (8)
Additional References                              343 (4)
  A References Added to the Text                   343 (2)
  B Further Reading for Chapter One                345 (1)
  C Further Reading for Chapter Two                345 (1)
  D Further Reading for Chapter Three              345 (1)
  E Further Reading for Chapter Four               346 (1)
Index                                              347