---
date: 2021-11-05 23:24
modification date: Friday 5th November 2021 23:24:58
title: absolute Galois group
aliases: [absolute Galois group]

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Last modified date: <%+ tp.file.last_modified_date() %>

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- Tags: 
	- #arithmetic-geometry 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

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# absolute Galois group

- Definition: For $k\in \Field$, the absolute Galois group is  
$$
G_k \da \Gal(k_s/k) \cong \Aut(\bar k/k)
.$$
  - Warning: $\bar k/k$ may not be Galois! Hence the need for a [separable closure](Unsorted/separable.md). 
- #why-care: $\Gal(\QQbar/\QQ)$ generalizes [class field theory](class%20field%20theory.md) and packages together all finite extensions of $\QQ$.
- For $K^\ab$ the [maximal abelian extension](maximal%20abelian%20extension) of $K$, finite abelian extensions of $K$ correspond to open subgroups of $\Gal(K^\ab/K)$, which are finite index since this group is compact.

![](attachments/Pasted%20image%2020220209174054.png)

![](attachments/Pasted%20image%2020220417012346.png)