Ring class field

In algebraic number theory, 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 $K$.

Examples

• Let $K$ be any number field. The ring class field for the maximal order ${𝒪}={{𝒪}}_K$ is the Hilbert class field $H$.

• Let $K=\mathbb{\mathbb{Q}}(\sqrt{-n})$.
  • The ring class field for the order ${𝒪}=\mathbb{\mathbb{Z}}[\sqrt{-n}]$ is $L=K(a)$, where $a$ is an algebraic integer with minimal polynomial over $\mathbb{\mathbb{Q}}$ of degree $h(-4n)$, the class number of an order with discriminant $-4n$. Moreover, if $p$ is an odd prime not dividing $n$, then $p$ splits completely in $L$ if and only if $p$ splits completely in $K$.

• • If ${𝒪}$ is an order and $a$ is a proper fractional O-ideal; i.e.

$\{x\in K^{\times }:xa\subset a\}={𝒪}$,

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

• "Varieties over special fields". Handbook of Elliptic and Hyperelliptic Curve Cryptography. Boca Raton, FL: Chapman & Hall. 2006. pp. 87–113. ISBN 978-1-58488-518-4. 

External links

• Osserman, Brian, Ring class fields and p = x² + ny² , https://web.archive.org/web/20180927172043/http://www.math.ucdavis.edu/~osserman/classes/254a/lectures/36.pdf, retrieved 5 August 2026