Finite Galois algebra

Let $F$ be a field and $G$ a finite group. A finite $G$ -Galois $F$ -algebra is a finite étale $F$ -algebra $A$ with an action of $G$ by $F$ -algebra automorphisms such that the canonical map

A\otimes_F A\longrightarrow\prod_{g\in G}A, \qquad a\otimes b\longmapsto\bigl(a\,g(b)\bigr)_{g\in G}

is an isomorphism.

Interpretation

Geometrically, $\operatorname{Spec}A\to\operatorname{Spec}F$ is a torsor under the constant finite group scheme attached to $G$ . If $\operatorname{Spec}A$ is connected, then $A$ is a field and $A/F$ is a finite Galois extension with group $G$ .

Remarks