Describe the Galois correspondence for a finite Galois extension $L / K$.

Let $L$ be the splitting field of $X^{4}-2$ over $\mathbb{Q}$. Compute the Galois group $G$ of $L / \mathbb{Q}$. For each subgroup of $G$, determine the corresponding subfield of $L$.

Let $L / K$ be a finite Galois extension whose Galois group is isomorphic to $S_{n}$. Show that $L$ is the splitting field of a separable polynomial of degree $n$.