Let $L=K\left(\xi_{n}\right)$, where $\xi_{n}$ is a primitive $n$th root of unity and $G=\operatorname{Aut}(L / K)$. Prove that there is an injective group homomorphism $\chi: G \rightarrow(\mathbb{Z} / n \mathbb{Z})^{*}$.

Show that, if $M$ is an intermediate subfield of $K\left(\xi_{n}\right) / K$, then $M / K$ is Galois. State carefully any results that you use.

Give an example where $G$ is non-trivial but $\chi$ is not surjective. Show that $\chi$ is surjective when $K=\mathbb{Q}$ and $n$ is a prime.

Determine all the intermediate subfields $M$ of $\mathbb{Q}\left(\xi_{7}\right)$ and the automorphism groups $\operatorname{Aut}\left(\mathbb{Q}\left(\xi_{7}\right) / M\right)$. Write the quadratic subfield in the form $\mathbb{Q}(\sqrt{d})$ for some $d \in \mathbb{Q}$.