(i) Let $K$ be the splitting field of the polynomial $f=X^{3}-2$ over the rationals. Find the Galois group $G$ of $K / \mathbb{Q}$ and describe its action on the roots of $f$.

(ii) Let $K$ be the splitting field of the polynomial $X^{4}+a X^{2}+b$ (where $a, b \in \mathbb{Q}$ ) over the rationals. Assuming that the polynomial is irreducible, prove that the Galois group $G$ of the extension $K / \mathbb{Q}$ is either $C_{4}$, or $C_{2} \times C_{2}$, or the dihedral group $D_{8}$.