Galois connection between subgroups and subfields

Definition

Let $L/K$ be a field extension, and let $G = \Aut(L/K)$ be the group of field automorphisms of $L$ that fix every element of $K$. Let $\mathcal{S}$ be the set of subgroups of $G$ and let $\mathcal{F}$ be the set of intermediate fields of $L/K$, both ordered by inclusion. For $H \in \mathcal{S}$, let $L^H = \set{x \in L : \sigma(x) = x \text{ for all } \sigma \in H}$ be the fixed field of $H$. For $E \in \mathcal{F}$, let $\Aut(L/E) = \set{\sigma \in G : \sigma(x) = x \text{ for all } x \in E}$. The Galois connection between subgroups and subfields of $L/K$ is the pair of maps $$\mathcal{S} \to \mathcal{F},\ H \mapsto L^H \qquad \text{and} \qquad \mathcal{F} \to \mathcal{S},\ E \mapsto \Aut(L/E).$$ It is an antitone Galois connection: for all $H \in \mathcal{S}$ and $E \in \mathcal{F}$, $$E \subseteq L^H \iff H \subseteq \Aut(L/E).$$