Wikiwand AI

Ring class field

From Wikipedia, the free encyclopedia

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 $K$ .^[1]

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 $K=\mathbb {Q} ({\sqrt {-n}})$ $K=\mathbb {Q} ({\sqrt {-n}})$ .
- 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 $-4n$ .^{[clarification needed]} 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.

Notes

References

External links

Related Articles

Timelines

Top Qs

Fact Checks