PlanetMath: ray class group

(more info)

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ray class group

(Definition)

Let ${\mathfrak{m}}$ be a modulus for a number field . The ray class group of mod ${\mathfrak{m}}$ is the group $\mathbb{I}^{\mathfrak{m}}/K_{{\mathfrak{m}},1}$ , where

$\mathbb{I}^{\mathfrak{m}}$ is the subgroup of the ideal group of generated by all prime ideals which do not occur in the factorization of ${\mathfrak{m}}$ .
$K_{{\mathfrak{m}},1}$ is the subgroup of $\mathbb{I}^{\mathfrak{m}}$ consisting of all principal ideals in the ring of integers of having the form $(\alpha)$ where $\alpha$ is multiplicatively congruent to $1 \bmod {\mathfrak{m}}$ .