# Galois groups and groups of Dirichlet characters

Assume we have Galois extension $K/\mathbb Q.$ Let be $(G,\circ)=\text{Gal}(K/\mathbb Q)$ its Galois group and let be $(X,+)$ group of Dirichlet characters, that "belongs to this field extension". Now, what is relation between $X$ and $G$?

• Could you make it more precise how Dirichlet characters are associated to $K/\Bbb{Q}$? For abelian extensions see here. – Dietrich Burde May 7 '18 at 11:08
• Sorry, I should have been more precise and specific. Let's start with a character $\chi\text{ mod }n:\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})\to\mathbb{C}^\times$, where I equate $\text{Gal}(\mathbb{Q}(\zeta_n)/\mathbb{Q})=(\mathbb{Z}/n\mathbb{Z})^\times$. Now can I somehow (in a different way) describe fixed field $F$ of $\ker \chi$? – byk7 May 7 '18 at 13:50