Ring class field

In algebraic number theory, a ring class field is the abelian extension of an algebraic number field ${\displaystyle K}$ associated by class field theory to the ring class group of some order ${\displaystyle {\mathcal {O}}}$ of the ring of integers of ${\displaystyle K}$.^[1]

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

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

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

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

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

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

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