Galois extension as an étale torsor

Statement

Let $K/F$ be a finite Galois extension, let $G=\operatorname{Gal}(K/F)$ , and set $P=\operatorname{Spec}K$ and $X=\operatorname{Spec}F$ . 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

P\times_X G_X\longrightarrow P\times_X P, \qquad (p,g)\longmapsto(p,p\cdot g)

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 $G$ .

Interpretation

The field inclusion $F\hookrightarrow K$ reverses direction under $\operatorname{Spec}$ , giving $P=\operatorname{Spec}K\to X=\operatorname{Spec}F$ . The theorem is the precise form of the slogan “a Galois extension is a principal bundle.”

Warning

The torsor is locally trivial in the étale topology, not generally in the Zariski topology. Calling it a “principal bundle” does not turn $\operatorname{Spec}K\to\operatorname{Spec}F$ into an ordinary smooth or topological bundle.