Let $K$ be a field of characteristic 0 containing all roots of unity.

(i) Let $L$ be the splitting field of the polynomial $X^{n}-a$ where $a \in K$. Show that the Galois group of $L / K$ is cyclic.

(ii) Suppose that $M / K$ is a cyclic extension of degree $m$ over $K$. Let $g$ be a generator of the Galois group and $\zeta \in K$ a primitive $m$-th root of 1 . By considering the resolvent

$$R(w)=\sum_{i=0}^{m-1} \frac{g^{i}(w)}{\zeta^{i}}$$

of elements $w \in M$, show that $M$ is the splitting field of a polynomial $X^{m}-a$ for some $a \in K$.