---
tag: F3GTF5NT4M
kind: concept
name: Krull topology
msc: [12F10, "20E18"]
sources:
  - url: https://en.wikipedia.org/wiki/Profinite_group
---

## Summary

The Krull topology is the natural topology on the [[Galois group|0XVR72M51M]] of a Galois extension, possibly of infinite degree.
A basis of neighbourhoods of the identity is given by the subgroups that fix a finite subextension.
With it, the Galois group is a [[profinite group|KA0YDB6QXE]], and [[infinite Galois theory|RVR5QFFD77]] matches intermediate fields with closed subgroups.

## Explanations

### L1 human

Two symmetries of a large field are declared close when they agree on a big finite piece of the field.
The more elements they agree on, the closer they are.
For a finite extension, every subgroup is open and the topology carries no new information.
For the algebraic closure of $\Q$, the topology is what makes the group of symmetries behave like a limit of finite groups.