Étale fundamental group

Summary

The étale fundamental group of a connected scheme classifies its finite étale covers, as the fundamental group of a space classifies its covering spaces. It is a profinite group. For the spectrum of a field it is the absolute Galois group. For a complex variety it is the profinite completion of the topological fundamental group. It unites Galois theory and the theory of coverings in one framework, Grothendieck's Galois theory.