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.^[1]

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[√−n] in the number field K = Q(√−n).

• 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(a) for a an algebraic integer with minimal polynomial over Q of degree h(−4n), the class number of an order with discriminant −4n.
• If O is an order and a is a proper fractional O-ideal (i.e. {x ϵ K* : xa ⊂ a} = O), write j(a) for the j-invariant of the associated elliptic curve. Then K(j(a)) is the ring class field of O and j(a) is an algebraic integer.