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.

Frey, Gerhard; Lange, Tanja (2006), "Varieties over special fields", Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math. Appl. (Boca Raton), Chapman & Hall/CRC, Boca Raton, Florida, pp. 87–113, MR 2162721. See in particular p. 99.

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