Étale algebra

In commutative algebra, an étale algebra over a field is a special type of algebra, one that is isomorphic to a finite product of finite separable field extensions. An étale algebra is a special sort of commutative separable algebra.

Definitions
Let $K$ be a field. Let $L$ be a commutative unital associative $K$-algebra. Then $L$ is called an étale $K$-algebra if any one of the following equivalent conditions holds: • $L\otimes_{K} E\simeq E^n$ for some field extension $E$ of $K$ and some nonnegative integer $n$.

• $L\otimes_{K} \overline{K} \simeq \overline{K}^n$ for any algebraic closure $\overline{K}$ of $K$ and some nonnegative integer $n$.

• $L$ is isomorphic to a finite product of finite separable field extensions of $K$.

• $L$ is finite-dimensional over $K$, and the trace form $Tr(xy)$ is nondegenerate.

• The morphism of schemes $\operatorname{Spec} L \to \operatorname{Spec} K$ is an étale morphism.

Examples
The $$\mathbb{Q}$$-algebra $$\mathbb{Q}(i)$$ is étale because it is a finite separable field extension.

The $$\mathbb{R}$$-algebra $$\mathbb{R}[x]/(x^2)$$ is not étale, since $$\mathbb{R}[x]/(x^2)\otimes_\mathbb{R}\mathbb{C} \simeq \mathbb{C}[x]/(x^2)$$.

Properties
Let $G$ denote the absolute Galois group of $K$. Then the category of étale $K$-algebras is equivalent to the category of finite $G$-sets with continuous $G$-action. In particular, étale algebras of dimension $n$ are classified by conjugacy classes of continuous homomorphisms from $G$ to the symmetric group $S_{n}$. These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.