Ring class field

In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.

Properties

Let K be an algebraic number field.

The ring class field for the maximal order O = O_K is the Hilbert class field H.

Let L be the ring class field for the order Z in the number field K = Q.

If p is an odd prime not dividing n, then p splits completely in L if and only if p splits completely in K.L = K for a an algebraic integer with minimal polynomial over Q of degree h, the class number of an order with discriminant −4n.
If O is an order and a is a proper fractional O-ideal, write j for the j-invariant of the associated elliptic curve. Then K is the ring class field of O and j is an algebraic integer.