# ray class group

Let ${\mathfrak{m}}$ be a modulus for a number field $K$. The ray class group of $K$ 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 $K$ 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 $K$ having the form $(\alpha)$ where $\alpha$ is multiplicatively congruent to $1\bmod{\mathfrak{m}}$.

Title ray class group RayClassGroup 2013-03-22 12:50:19 2013-03-22 12:50:19 djao (24) djao (24) 4 djao (24) Definition msc 11R29