Dirichlet group of an Abelian number field (reviewed)

If $K$ is an abelian number field, then $K\subseteq \Q(\zeta_n)$ for some positive integer $n$. Take the minimal such $n$, i.e., the conductor of $K$.

The Galois group $\Gal(\Q(\zeta_n)/\Q)$ is canonically isomorphic to $\Z_n^\times$. The Dirichlet characters modulo $n$ form the dual group of homomorphisms $\chi:\Z_n^\times\to\C^\times$. Since $\Gal(K/\Q)$ is a quotient group of $\Gal(\Q(\zeta_n)/\Q)$, its dual group is a subgroup of the group of Dirichlet characters modulo $n$, called the Dirichlet character group of $K$.