Ray Class Structure and Fields, Hilbert Class Fields

Abstract

The concepts of strict ideal classes, ring ideal theory, and genus theory for quadratic forms generalize for an arbitrary field k (⊇ℚ) into what we call “ray” ideal classes. This is an intrinsic property of k. It corresponds to an extrinsic property of k, its extension fields K/k. What concerns us here is even more special, it is the manner in which a prime ideal p in k will factor in K, say as

$$ p = p_1^{{e_1}} \cdots \mathop p\limits_g^e g. $$

We ask two questions:

• (i) Which primes p lead to an exponent e_i> 1, i.e., which p ramify over (one or more) P_i? Such primes, if they occur, are (later) seen to be finite in number for any K/k.

• (ii) Which primes p (completely) split, or which factor so that each e_l = 1 and g = [K: k]? There are seen to be infinitely many such p (although we must wait for Chapter 19 to see this).