(i) Let $K$ be the splitting field of the polynomial

$$x^{4}-4 x^{2}-1$$

over $\mathbb{Q}$. Show that $[K: \mathbb{Q}]=8$, and hence show that the Galois group of $K / \mathbb{Q}$ is the dihedral group of order 8 .

(ii) Let $L$ be the splitting field of the polynomial

$$x^{4}-4 x^{2}+1$$

over $\mathbb{Q}$. Show that $[L: \mathbb{Q}]=4$. Show that the Galois group of $L / \mathbb{Q}$ is $C_{2} \times C_{2}$.