Title Page ; Copyright ; Contents ; Preface to the First Edition ; Preface to the Second Edition ; Notation ; Introduction ; Chapter One: From Fermat to Gauss ; 1. Fermat, Euler and Quadratic Reciprocity ; A. Fermat ; B. Euler ; C.P = x2 + ny2 and Quadratic Reciprocity ; D. Beyond Quadratic Reciprocity ; E. Exercises ; 2. Lagrange, Legendre and Quadratic Forms ; A. Quadratic Forms ; B.P = x2 + ny2 and Quadratic Forms ; C. Elementary Genus Theory ; D. Lagrange and Legendre ; E. Exercises ; 3. Gauss, Composition and Genera ; A. Composition and the Class Group ; B. Genus Theory. C.P = x2 + ny2 and Euler''s Convenient Numbers D. Disquisitiones Arithmeticae ; E. Exercises ; 4. Cubic and Biquadratic Reciprocity ; A. Z[w] and Cubic Reciprocity ; B. Z[i] and Biquadratic Reciprocity ; C. Gauss and Higher Reciprocity ; D. Exercises ; Chapter Two: Class Field Theory ; 5. The Hilbert Class Field and P = x2 + ny2 ; A. Number Fields ; B. Quadratic Fields ; C. The Hilbert Class Field ; D. Solution of P = x2 + ny2 for Infinitely Many n ; E. Exercises ; 6. The Hilbert Class Field and Genus Theory ; A. Genus Theory for Field Discriminants. B. Applications to the Hilbert Class Field 7. Orders in Imaginary Quadratic Fields ; A. Orders in Quadratic Fields ; B. Orders and Quadratic Forms ; C. Ideals Prime to the Conductor ; D. The Class Number ; E. Exercises ; 8. Class Field Theory and the Cebotarev Density Theorem ; A. The Theorems of Class Field Theory ; B. The Čebotarev Density Theorem ; C. Norms and Ideles ; D. Exercises ; 9. Ring Class Fields and p = x2 + ny2 ; A. Solution of p = x2 + ny2 for All n ; B. The Ring Class Fields of Z[[square root]-27] and Z[[square root]-64] ; C. Primes Represented by Positive Definite Quadratic Forms. B. The Weber Functions C. J-invariants of Orders of Class Number 1 ; D. Weber''s Computation of J ([square root]-14) ; E. Imaginary Quadratic Fields of Class Number 1 ; F. Exercises ; 13. The Class Equation ; A. Computing the Class Equation ; B. Computing the Modular Equation ; C. Theorems of Deuring, Gross and Zagier ; D. Exercises ; Chapter Four: Additional Topics ; 14. Elliptic Curves ; A. Elliptic Curves and Weierstrass Equations ; B. Complex Multiplication and Elliptic Curves ; C. Elliptic Curves over Finite Fields ; D. Elliptic Curve Primality Tests ; E. Exercises ; 15. Shimura Reciprocity.

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 ReviewsWritten 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 fi.