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  "body": "\n## Definition\n\nLet $k$ be a [[field|0QA8CHH8MG]] and let $k^s$ be a [[separable closure|JJ91A1V8NQ]] of $k$.\nLet $G = \\Gal(k^s/k)$ be the [[absolute Galois group|PGKWBJDTKX]] with its [[Krull topology|F3GTF5NT4M]].\nA *finite étale $k$-algebra* is a $k$-algebra isomorphic to a finite product $L_1 \\times \\cdots \\times L_r$.\nHere each $L_i$ is a finite [[separable extension|0N019F6RE9]] of $k$.\nFor 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|0YSE802YCQ]] $g \\cdot \\varphi = g \\circ \\varphi$ of $G$.\nLet $\\mathcal{E}_k$ be the category of finite étale $k$-algebras, and let $\\mathcal{S}_G$ be the category of finite sets with a continuous action of $G$.\n*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.\n\n## Remarks\n\nUnder $F$, the dimension $\\dim_k A$ equals the number of elements of $F(A)$.\nA field $L$ goes to a transitive $G$-set, and a product of algebras goes to a disjoint union of $G$-sets.\nFor $k = \\R$, the group $G$ has order $2$, and $F(\\C)$ is a set of two points swapped by $G$, while $F(\\R \\times \\R)$ is a set of two fixed points.\nThe algebra $k[x]/(x^2)$ is not finite étale, because it has a nonzero nilpotent element, and no product of fields has one.\nThe general form uses a Galois category with a fibre functor; for a connected scheme it defines the étale fundamental group, and for $\\operatorname{Spec} k$ this group is $G$.\n",
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