---
tag: RVR5QFFD77
kind: concept
name: Infinite Galois theory
msc: [12F10, "20E18"]
specializes: [WNQDKQYZ2R]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_extension
---

## Summary

Infinite Galois theory extends [[Galois theory|WNQDKQYZ2R]] to [[Galois extensions|0DS2F7GK05]] of infinite degree, such as the algebraic closure of $\Q$ over $\Q$.
The Galois group then carries the [[Krull topology|F3GTF5NT4M]], which makes it a [[profinite group|KA0YDB6QXE]].
The fundamental theorem survives with one change: intermediate fields correspond to the closed subgroups, not to all subgroups.

## Explanations

### L1 human

An infinite Galois extension is the union of its finite Galois pieces.
Each symmetry of the whole extension is a compatible choice of symmetries of all these pieces.
Two symmetries are close in the Krull topology when they agree on a large finite piece.
For example, the symmetries of the algebraic closure of $\F_p$ include the Frobenius map $x \mapsto x^p$, and the closed subgroup it generates is the whole group.