Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 1 / e / Solution

Since $5$ is inert, $\mathfrak{m}=5{\mathcal{O}}_K$ is prime andThe unit group is $\{\pm 1\}$, and only $1$ is congruent to $1$ modulo $\mathfrak{m}$, soThere are no real places and $h_K=1$. The ray class number formula givesHence the ray class field modulo $5{\mathcal{O}}_K$ has degree $\boxed{12}$ over $K$.