Grothendieck's Galois theory

Definition

Let $k$ be a field and let $k^s$ be a separable closure of $k$. Let $G = \Gal(k^s/k)$ be the absolute Galois group with its Krull topology. A finite étale $k$-algebra is a $k$-algebra isomorphic to a finite product $L_1 \times \cdots \times L_r$. Here each $L_i$ is a finite separable extension of $k$. For such an algebra $A$, let $F(A) = \Hom_k(A, k^s)$ be the set of $k$-algebra homomorphisms $A \to k^s$, with the action $g \cdot \varphi = g \circ \varphi$ of $G$. Let $\mathcal{E}_k$ be the category of finite étale $k$-algebras, and let $\mathcal{S}_G$ be the category of finite sets with a continuous action of $G$. Grothendieck's Galois theory for $k$ is the statement that $F \colon \mathcal{E}_k \to \mathcal{S}_G$ is an anti-equivalence of categories.