---
tag: KJ11DAXMR0
kind: concept
name: Étale fundamental group
msc: [14F35, "14E20"]
sources:
  - url: https://en.wikipedia.org/wiki/%C3%89tale_fundamental_group
same_as:
  wikidata: Q288764
  wikipedia: Étale fundamental group
---

## Summary

The étale fundamental group of a connected scheme classifies its finite étale covers, as the [[fundamental group|S1XK3HXHFX]] of a space classifies its [[covering spaces|3XZRRE165N]].
It is a [[profinite group|KA0YDB6QXE]].
For the spectrum of a field it is the [[absolute Galois group|DPFMXS71D1]].
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|CX7BDQ47ZK]].

## Explanations

### L1 human

A loop in a space can be detected by the covers it produces: going around the loop permutes the sheets of a cover.
In algebraic geometry one cannot draw loops, but one can still study finite covers.
The étale fundamental group is the group of symmetries of all finite covers at once.
For the punctured complex line, the finite covers are $z \mapsto z^n$, and the group is the profinite completion $\widehat{\Z}$ of $\Z$.
For a field $k$, the finite covers are the finite separable field extensions, so the group is the Galois group of the separable closure.