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

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