Primes of the Form $x^2 + Ny^2$ : Fermat, Class Field Theory, and Complex Multiplication. Third Edition with Solutions (Chelsea Publications)

Cox, David A.

American Mathematical Society（2022/12発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 542 p.
• 言語 ENG
• 商品コード 9781470470289
• DDC分類 512.723

Full Description

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

From Fermat to Gauss
Class field theory
Complex multiplication
Additional topics
Solutions
References
Further reading
Index