Let $M / K$ be a finite Galois extension of fields. Explain what is meant by the Galois correspondence between subfields of $M$ containing $K$ and subgroups of $\operatorname{Gal}(M / K)$. Show that if $K \subset L \subset M$ then $\operatorname{Gal}(M / L)$ is a normal subgroup of $\operatorname{Gal}(M / K)$ if and only if $L / K$ is normal. What is $\operatorname{Gal}(L / K)$ in this case?

Let $M$ be the splitting field of $X^{4}-3$ over $\mathbb{Q}$. Prove that $\operatorname{Gal}(M / \mathbb{Q})$ is isomorphic to the dihedral group of order 8. Hence determine all subfields of $M$, expressing each in the form $\mathbb{Q}(x)$ for suitable $x \in M$.