Finite Galois algebra - Imported Knowlpedia Content

Definition

A finite Galois field extension is the connected case. If connectedness is dropped, the same symmetry can act transitively across several field factors, so the correct algebraic object is more general than a field.

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. Equivalently, $A^G=F$ and $\dim_F A=|G|$ , together with the corresponding Galois descent condition.

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 $G$ .