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 = OK is the Hilbert class field H.
Let L be the ring class field for the order Z 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.
References
- 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.
External links
- Ring class fields. Archived 27 September 2018 at the Wayback Machine
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