(a) Let $K$ be a field and let $L$ be the splitting field of a polynomial $f(x) \in K[x]$. Let $\xi_{N}$ be a primitive $N^{\text {th }}$root of unity. Show that $\operatorname{Aut}\left(L\left(\xi_{N}\right) / K\left(\xi_{N}\right)\right)$ is a subgroup of $\operatorname{Aut}(L / K)$.

(b) Suppose that $L / K$ is a Galois extension of fields with cyclic Galois group generated by an element $\sigma$ of order $d$, and that $K$ contains a primitive $d^{\text {th }}$root of unity $\xi_{d}$. Show that an eigenvector $\alpha$ for $\sigma$ on $L$ with eigenvalue $\xi_{d}$ generates $L / K$, that is, $L=K(\alpha)$. Show that $\alpha^{d} \in K$.

(c) Let $G$ be a finite group. Define what it means for $G$ to be solvable.

Determine whether

(i) $G=S_{4} ; \quad$ (ii) $G=S_{5}$

are solvable.

(d) Let $K=\mathbb{Q}\left(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)$ be the field of fractions of the polynomial ring $\mathbb{Q}\left[a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right]$. Let $f(x)=x^{5}-a_{1} x^{4}+a_{2} x^{3}-a_{3} x^{2}+a_{4} x-a_{5} \in K[x]$. Show that $f$ is not solvable by radicals. [You may use results from the course provided that you state them clearly.]