Étale algebra
Introduction 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  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:
Examples The -algebra  is étale because it is a finite separable field extension.The -algebra  is not étale, since .
Properties 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 homomorphisms from  to the symmetric group . These globalize to e.g. the definition of étale fundamental groups and categorify to Grothendieck's Galois theory.
Notes
References ** http://www.jmilne.org/math/CourseNotes/FT.pdf