Let $n \geqslant 1$ and $K=\mathbb{Q}\left(\boldsymbol{\mu}_{n}\right)$ be the cyclotomic field generated by the $n$th roots of unity. Let $a \in \mathbb{Q}$ with $a \neq 0$, and consider $F=K(\sqrt[n]{a})$.

(i) State, without proof, the theorem which determines $\operatorname{Gal}(K / \mathbb{Q})$.

(ii) Show that $F / \mathbb{Q}$ is a Galois extension and that $\operatorname{Gal}(F / \mathbb{Q})$ is soluble. [When using facts about general Galois extensions and their generators, you should state them clearly.]

(iii) When $n=p$ is prime, list all possible degrees $[F: \mathbb{Q}]$, with justification.