Let $f$ be a separable polynomial of degree $n \geqslant 1$ over a field $K$. Explain what is meant by the Galois group $\operatorname{Gal}(f / K)$ of $f$ over $K$. Explain how $\operatorname{Gal}(f / K)$ can be identified with a subgroup of the symmetric group $S_{n}$. Show that as a permutation group, $\operatorname{Gal}(f / K)$ is transitive if and only if $f$ is irreducible over $K$.

Show that the Galois group of $f(X)=X^{5}+20 X^{2}-2$ over $\mathbb{Q}$ is $S_{5}$, stating clearly any general results you use.

Now let $K / \mathbb{Q}$ be a finite extension of prime degree $p>5$. By considering the degrees of the splitting fields of $f$ over $K$ and $\mathbb{Q}$, show that $\operatorname{Gal}(f / K)=S_{5}$ also.