Galois correspondence theorem

Let $K \subseteq L$ a finite Galois extension with Galois group $G=\operatorname{Gal}(L/K)$.
Then, the map sending a subgroup $H$ to its associated fixed field $L^{H}$ is an order-reversing bijection between the set of subgroups of $G$ and the intermediate fields $K \subseteq E\subseteq L$, whose inverse sends each intermediate field $E$ to its fixing subgroup.

Examples

Proof