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.
Let be a field. Let be a commutative unital associative -algebra. Then is called an étale -algebra if any one of the following equivalent conditions holds:
The -algebra is étale because it is a finite separable field extension.
The -algebra of dual numbers is not étale, since .
Let denote the absolute Galois group of . Then the category of étale -algebras is equivalent to the category of finite -sets with continuous -action. In particular, étale algebras of dimension are classified by conjugacy classes of continuous group homomorphisms from to the symmetric group . These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.