(a) Let $L$ be the 13 th cyclotomic extension of $\mathbb{Q}$, and let $\mu$ be a 13 th primitive root of unity. What is the minimal polynomial of $\mu$ over $\mathbb{Q}$ ? What is the Galois group $\operatorname{Gal}(L / \mathbb{Q})$ ? Put $\lambda=\mu+\frac{1}{\mu}$. Show that $\mathbb{Q} \subseteq \mathbb{Q}(\lambda)$ is a Galois extension and find $\operatorname{Gal}(\mathbb{Q}(\lambda) / \mathbb{Q})$.

(b) Define what is meant by a Kummer extension. Let $K$ be a field of characteristic zero and let $L$ be the $n$th cyclotomic extension of $K$. Show that there is a sequence of Kummer extensions $K=F_{1} \subseteq F_{2} \subseteq \cdots \subseteq F_{r}$ such that $L$ is contained in $F_{r}$.