Primes of the Form x2+ny2: Fermat, Class Field Theory, and Complex Multiplication

PREFACE TO THE FIRST EDITION

Several years ago, while reading Weil's Number Theory: An Approach Through History, I noticed a conjecture of Euler concerning primes of the form x² + 14y². That same week I picked up Cohn's A Classical Invitation to Algebraic Numbers and Class Fields and saw the same example treated from the point of view of the Hilbert class field. The coincidence made it clear that something interesting was going on, and this book is my attempt to tell the story of this wonderful part of mathematics.

I am an algebraic geometer by training, and number theory has always been more of an avocation than a profession for me. This will help explain some of the curious omissions in the book. There may also be errors of history or attribution (for which I take full responsibility), and doubtless some of the proofs can be improved. Corrections and comments are welcome!

I would like to thank my colleagues in the number theory seminars of Oklahoma State University and the Five Colleges (Amherst College, Hampshire College, Mount Holyoke College, Smith College and the University of Massachusetts) for the opportunity to present material from this book in preliminary form. Special thanks go to Dan Flath and Peter Norman for their comments on earlier versions of the manuscript. I also thank the reference librarians at Amherst College and Oklahoma Slate University for their help in obtaining books through interlibrary loan.

DAVID A. COX

Amherst, Massachsusetts

August 1989