---
tag: PGKWBJDTKX
kind: concept
name: Absolute Galois group
msc: [12F10, "20E18", 11R32]
specializes: [0XTY86BZKY]
sources:
  - url: https://en.wikipedia.org/wiki/Absolute_Galois_group
same_as:
  wikidata: Q332407
  wikipedia: Absolute Galois group
---

## Summary

The absolute Galois group of a field $K$ is the Galois group of a [[separable closure|JJ91A1V8NQ]] of $K$ over $K$.
It contains, as quotients, the Galois groups of all finite Galois extensions of $K$, so it records the whole Galois theory of $K$ at once.
With the [[Krull topology|F3GTF5NT4M]] it is a [[profinite group|KA0YDB6QXE]].

## Explanations

### L1 human

Instead of studying one extension of a field at a time, take the largest separable algebraic extension and look at all its symmetries.
Every finite Galois extension is a piece of it, and its group is a quotient of the absolute Galois group.
For the real numbers, the separable closure is $\C$, and the absolute Galois group has two elements: the identity and complex conjugation.
For $\Q$, the absolute Galois group is a huge and still mysterious object.