(i) Let $K$ be a field, $\theta \in K$, and $n>0$ not divisible by the characteristic. Suppose that $K$ contains a primitive $n$th root of unity. Show that the splitting field of $x^{n}-\theta$ has cyclic Galois group.

(ii) Let $L / K$ be a Galois extension of fields and $\zeta_{n}$ denote a primitive $n$th root of unity in some extension of $L$, where $n$ is not divisible by the characteristic. Show that $\operatorname{Aut}\left(L\left(\zeta_{n}\right) / K\left(\zeta_{n}\right)\right)$ is a subgroup of $\operatorname{Aut}(L / K)$.

(iii) Determine the minimal polynomial of a primitive 6 th root of unity $\zeta_{6}$ over $\mathbf{Q}$.

Compute the Galois group of $x^{6}+3 \in \mathbf{Q}[x]$.