---
tag: R3625YKQC0
kind: concept
name: Galois connection between subgroups and subfields
msc: [12F10, 06A15]
sources:
  - url: https://en.wikipedia.org/wiki/Galois_connection
---

## Summary

Let $L/K$ be a field extension.
One map sends a group $H$ of automorphisms of $L$ over $K$ to its fixed field.
The other sends an intermediate field $E$ to the group of automorphisms that fix $E$.
Together they form an antitone Galois connection.
Both maps reverse inclusions, and applying one after the other can only enlarge a subgroup or an intermediate field.
For a finite [[Galois extension|0DS2F7GK05]] the two maps are inverse bijections; this is the [[fundamental theorem of Galois theory|0T1Q8D0FW8]].

## Explanations

### L1 human

Think of two lists side by side: the subgroups of the automorphism group, and the fields between $K$ and $L$.
Each subgroup points to the field of elements it leaves fixed, and each field points to the group of automorphisms that leave it fixed.
A bigger group fixes fewer elements, so the arrows turn the order upside down.
For $L = \Q(\sqrt{2})$ over $K = \Q$, the trivial group points to $L$, and the group generated by $\sqrt{2} \mapsto -\sqrt{2}$ points to $\Q$.