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

CHAPTER TWO

CLASS FIELD THEORY

§5. THE HILBERT CLASS FIELD AND p = x² + ny²

In Chapter One, we used elementary techniques to study the primes represented by x² + ny², n > 0. Genus theory told us when p = x² + ny² for a large but finite number of n's, and cubic and biquadratic reciprocity enabled us to treat two cases where genus theory failed. These methods are lovely but limited in scope. To solve p = x² + ny² when n > 0 is arbitrary, we will need class field theory, and this is the main task of Chapter Two. But rather than go directly to the general theorems of class field theory, in §5 we will first study the special case of the Hilbert class field. Theorem 5.1 below will use Artin reciprocity for the Hilbert class field to solve our problem for infinitely many (but not all) n > 0. We will then study the case p = x² + 14y² in detail. This is a case where our previous methods failed, but once we determine the Hilbert class field of , Theorem 5.1 will immediately give us a criterion for when p = x² + 14y².

The central notion of this section is the Hilbert class field of a number field K. We do not assume any previous acquaintance with this topic, for one of our goals is to introduce the reader to this more accessible part of class field theory. To see what the Hilbert class field has to do with the problem of representing primes by x² + ny², let's state the main theorem we intend ...