Primes of the Form X^2 + Ny^2: Fermat, Class Field Theory, and Complex Multiplication, With Solutions (Ams Chelsea Publishing, 387)

This book studies when a prime $p$ can be written in the form $x^{2} + ny^{2}$. It begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. After exploring cubic and biquadratic reciprocity, the pace quickens with the introduction of algebraic number fields and class field theory. This leads to the concept of ring class field and a complete but abstract solution of $p = x^{2} + ny^{2}$. To make things more concrete, the book introduces complex multiplication and modular functions to give a constructive solution. The book ends with a discussion of elliptic curves and Shimura reciprocity. Along the way the reader will encounter some compelling history and marvelous formulas, together with a complete solution of the class number one problem for imaginary quadratic fields. The book is accessible to readers with modest backgrounds in number theory. In the third edition, the numerous exercises have been thoroughly checked and revised, and as a special feature, complete solutions are included. This makes the book especially attractive to readers who want to get an active knowledge of this wonderful part of mathematics. Contents Preface First Edition Second Edition Third Edition with Solutions Notation Introduction Chapter 1. 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[ω] and Cubic Reciprocity B. Z[i] and Biquadratic Reciprocity C. Gauss and Higher Reciprocity D. Exercises Chapter 2. 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 C. Exercises 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 Čebotarev 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[√-27] and Z[√-64] C. Primes Represented by Positive Definite Quadratic Forms D. Ring Class Fields and Generalized Dihedral Extensions E. Exercises Chapter 3. Complex Multiplication 10. Elliptic Functions and Complex Multiplication A. Elliptic Functions and the Weierstrass ℘-Function B. The j-Invariant of a Lattice C. Complex Multiplication D. Exercises 11. Modular Functions and Ring Class Fields A. The j-Function B. Modular Functions for Γ0(m) C. The Modular Equation Φm(X,Y) D. Complex Multiplication and Ring Class Fields E. Exercises 12. Modular Functions and Singular j-Invariants A. The Cube Root of the j-Function B. The Weber Functions C. j-Invariants of Orders of Class Number 1 D. Weber’s Computation of j(√-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 4. 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 A. Modular Functions B. The Shimura Reciprocity Theorem C. Extended Ring Class Fields D. Shimura Reciprocity for Extended Ring Class Fields E. Shimura Reciprocity for Ring Class Fields F. Class Field Theory G. Exercises Solutions by Roger Lipsett and David Cox Solutions to Exercises in §1 Solutions to Exercises in §2 Solutions to Exercises in §3 Solutions to Exercises in §4 Solutions to Exercises in §5 Solutions to Exercises in §6 Solutions to Exercises in §7 Solutions to Exercises in §8 Solutions to Exercises in §9 Solutions to Exercises in §10 Solutions to Exercises in §11 Solutions to Exercises in §12 Solutions to Exercises in §13 Solutions to Exercises in §14 Solutions to Exercises in §15 References Further Reading Index Examines when a prime $p$ can be written in the form $x^{2} + ny^{2}$. The book begins at an elementary level with results of Fermat and Euler and then discusses the work of Lagrange, Legendre and Gauss on quadratic reciprocity and the genus theory of quadratic forms. "The goal of the new edition of Primes of the Form x2 +ny2 is to make this wonderful part of number theory available to readers in a form especially suited to self-study, mainly because complete solutions to all exercises are included"-- Provided by publisher