---
tag: 218WAD58W3
kind: definition
name: Grothendieck's Galois theory
msc: [12F10, 14F35]
defines: CX7BDQ47ZK
uses: [0QA8CHH8MG, 0N019F6RE9, JJ91A1V8NQ, PGKWBJDTKX, F3GTF5NT4M, 0YSE802YCQ]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck%27s_Galois_theory
---

## Definition

Let $k$ be a [[field|0QA8CHH8MG]] and let $k^s$ be a [[separable closure|JJ91A1V8NQ]] of $k$.
Let $G = \Gal(k^s/k)$ be the [[absolute Galois group|PGKWBJDTKX]] with its [[Krull topology|F3GTF5NT4M]].
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|0N019F6RE9]] 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|0YSE802YCQ]] $g \cdot \varphi = g \circ \varphi$ 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.

## Remarks

Under $F$, the dimension $\dim_k A$ equals the number of elements of $F(A)$.
A field $L$ goes to a transitive $G$-set, and a product of algebras goes to a disjoint union of $G$-sets.
For $k = \R$, 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.
The algebra $k[x]/(x^2)$ is not finite étale, because it has a nonzero nilpotent element, and no product of fields has one.
The 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 $G$.