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 ${\mathcal {O}}$ of the ring of integers of $K$.

Examples

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

• Let $K=\mathbb {Q} ({\sqrt {-n}})$.
  • The ring class field for the order ${\mathcal {O}}=\mathbb {Z} [{\sqrt {-n}}]$ is $L=K(a)$, where $a$ is an algebraic integer with minimal polynomial over $\mathbb {Q}$ 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 ${\mathcal {O}}$ is an order and $a$ is a proper fractional O-ideal; i.e.

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

write $j(a)$ for the j-invariant of the associated elliptic curve. Then $K(j(a))$ is the ring class field of ${\mathcal {O}}$ and $j(a)$ is an algebraic integer.

References

• Frey, Gerhard; Lange, Tanja (2006). "Varieties over special fields". In Cohen, Henri; Frey, Gerhard (eds.). Handbook of Elliptic and Hyperelliptic Curve Cryptography. Boca Raton, FL: Chapman & Hall. pp. 87–113. ISBN 978-1-58488-518-4. MR 2162721.

External links

• Osserman, Brian, Ring class fields and p = x² + ny²  (PDF), retrieved 5 August 2026