Galois extension as an étale torsor

Statement

This is the precise form of the slogan “a Galois extension is a principal bundle.” The field inclusion $F\hookrightarrow K$ reverses direction under $\operatorname{Spec}$, producing

Let $K/F$ be a finite Galois extension and let $G=\operatorname{Gal}(K/F)$. Then $P\to X$ is finite étale and connected. The action of $G$ on $K$ by $F$-algebra automorphisms induces, contravariantly, an action on $P$. With $G_X$ the constant finite group scheme, the map

is an isomorphism. Hence $P\to X$ is a $G_X$-torsor on the étale site of $X$, in the sense of a $G$-torsor on a site. The algebra behind the isomorphism is the Galois tensor-product identity.

Conversely, a connected finite étale $G_X$-torsor over $\operatorname{Spec}F$ has affine total space $\operatorname{Spec}K$, where $K/F$ is a finite Galois extension with group $G$.