PlanetMath: ray class field

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ray class field

(Definition)

Proposition 1 Let be a finite abelian extension of number fields, and let $\mathcal{O}_K$ be the ring of integers of . There exists an integral ideal $\mathcal{C}\subset \mathcal{O}_K$ , divisible by precisely the prime ideals of that ramify in , such that

$\displaystyle \left((\alpha),L/K\right)=1,\quad \forall \alpha \in K^{\ast}, \alpha\equiv1\ \operatorname{mod}\ \mathcal{C}$

where $\left((\alpha),L/K\right)$ is the Artin map.

Definition 1 The conductor of a finite abelian extension is the largest ideal $\mathcal{C}_{L/K}\subset \mathcal{O}_K$ satisfying the above properties.

Note that there is a “largest ideal” with this condition because if proposition 1 is true for $\mathcal{C}_1,\mathcal{C}_2$ then it is also true for $\mathcal{C}_1+\mathcal{C}_2$ .

Definition 2 Let $\mathcal{I}$ be an integral ideal of . A ray class field of (modulo $\mathcal{I}$ ) is a finite abelian extension $K_{\mathcal{I}}/K$ with the property that for any other finite abelian extension with conductor $\mathcal{C}_{L/K}$ ,

$\displaystyle \mathcal{C}_{L/K} \mid \mathcal{I}\Rightarrow L\subset K_{\mathcal{I}}$

Note: It can be proved that there is a unique ray class field with a given conductor. In words, the ray class field is the biggest abelian extension of with a given conductor (although the conductor of $K_{\mathcal{I}}$ does not necessarily equal $\mathcal{I}$ !, see example ).

Remark: Let $\mathfrak{p}$ be a prime of unramified in , and let $\mathfrak{P}$ be a prime above $\mathfrak{p}$ . Then $(\mathfrak{p},L/K)=1$ if and only if the extension of residue fields is of degree 1

$\displaystyle [\mathcal{O}_L/\mathfrak{P}\colon \mathcal{O}_K/\mathfrak{p}]=1$

if and only if $\mathfrak{p}$ splits completely in

. Thus we obtain a characterization of the ray class field of conductor $\mathcal{C}$ as the abelian extension of

such that a prime of

splits completely if and only if it is of the form

$\displaystyle (\alpha),\quad \alpha \in K^{\ast}, \alpha\equiv1\ \operatorname{mod}\ \mathcal{C}$

Examples:

The ray class field of $\mathbb{Q}$ of conductor $N\mathbb{Z}$ is the $N^{th}$ -cyclotomic extension of $\mathbb{Q}$ . More concretely, let $\zeta_N$ be a primitive $N^{th}$ root of unity. Then
$\displaystyle \mathbb{Q}_{N\mathbb{Z}}=\mathbb{Q}(\zeta_N)$
$\displaystyle \mathbb{Q}(i)_{(2)}=\mathbb{Q}(i)$
so the conductor of $\mathbb{Q}(i)_{(2)}/\mathbb{Q}$ is .
$K_{(1)}$ , the ray class field of conductor , is the maximal abelian extension of which is unramified everywhere. It is, in fact, the Hilbert class field of .