Let $K$ be a field, and my definition of étale $K$-algebra is $K$-algebra $L$ which is isomorphic to a finite product of finite separable field extensions of $K$.
I found
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 properties section of this wikipedia page, and want to show this by checking that it satisfies the definition of the Galois category by Grothendieck.
First of all, I searched reference of galois category and there are two different definition. The one that does not use strict epimorphism is in this pdf, and the second is use strict epimorphism is in this pdf.
I have two questions.
- Which is good for beginner of galois theory? Or they are equivalent?
- What are morphisms of the category of étale $K$-algebras? Are they $K$-algebra morphisms?