Let $f \in \mathbb{Q}[t]$ be of degree $n>0$, with no repeated roots, and let $L$ be a splitting field for $f$.

(i) Show that $f$ is irreducible if and only if for any $\alpha, \beta \in \operatorname{Root}_{f}(L)$ there is $\phi \in \operatorname{Gal}(L / \mathbb{Q})$ such that $\phi(\alpha)=\beta$.

(ii) Explain how to define an injective homomorphism $\tau: \operatorname{Gal}(L / \mathbb{Q}) \rightarrow S_{n}$. Find an example in which the image of $\tau$ is the subgroup of $S_{3}$ generated by (2 3). Find another example in which $\tau$ is an isomorphism onto $S_{3}$.

(iii) Let $f(t)=t^{5}-3$ and assume $f$ is irreducible. Find a chain of subgroups of $\operatorname{Gal}(L / \mathbb{Q})$ that shows it is a solvable group. [You may quote without proof any theorems from the course, provided you state them clearly.]