Grothendieck's Galois theory

Summary

Grothendieck's Galois theory recasts Galois theory as an equivalence of categories. For a field $k$, it matches finite étale $k$-algebras with finite sets on which the absolute Galois group of $k$ acts continuously. The same pattern, with Galois categories and fibre functors, defines the étale fundamental group of a scheme. It thus puts Galois groups of fields and fundamental groups of spaces in one frame.