---
tag: CX7BDQ47ZK
kind: concept
name: Grothendieck's Galois theory
msc: [12F10, 14F35]
sources:
  - url: https://en.wikipedia.org/wiki/Grothendieck%27s_Galois_theory
same_as:
  wikidata: Q5610719
  wikipedia: Grothendieck's Galois theory
---

## Summary

Grothendieck's Galois theory recasts [[Galois theory|WNQDKQYZ2R]] as an equivalence of categories.
For a field $k$, it matches finite étale $k$-algebras with finite sets on which the [[absolute Galois group|PGKWBJDTKX]] 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.

## Explanations

### L1 human

Classical Galois theory matches intermediate fields with subgroups of a Galois group.
Grothendieck's view looks instead at all finite separable algebras at once, such as products of fields.
Each such algebra is replaced by the finite set of its embeddings into a separable closure, and the Galois group permutes this set.
For $k = \R$, the algebra $\C$ gives a set of two points swapped by complex conjugation, while $\R \times \R$ gives two points that stay fixed.
The picture is the same as for covering spaces, where the fundamental group permutes the points of a fibre.