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

INTRODUCTION

Most first courses in number theory or abstract algebra prove a theorem of Fermat which states that for an odd prime p,

This is only the first of many related results that appear in Fermat's works. For example, Fermat also states that if p is an odd prime, then

These facts are lovely in their own right, but they also make one curious to know what happens for primes of the form x² + 5y², x² + 6y², etc. This leads to the basic question of the whole book, which we formulate as follows:

Basic Question 0.1. Given a positive integer n, which primes p can be expressed in the form

where x and y are integers?

We will answer this question completely, and along the way we will encounter some remarkably rich areas of number theory. The first steps will be easy, involving only quadratic reciprocity and the elementary theory of quadratic forms in two variables over . These methods work nicely in the special cases considered above by Fermat. Using genus theory and cubic and biquadratic reciprocity, we can treat some more cases, but elementary methods fail to solve the problem in general. ...