---
type: "concept"
title: "Galois correspondence theorem"
slug: "galois-correspondence-theorem"
language: "en"
translationGroupId: "cmramu5xu0009pl01m31qdk62"
domain: "Algebra"
status: "stub"
aliases: ["Fundamental theorem of Galois theory"]
lastEditedBy: "ancient-tree"
---

Let $K \subseteq L$ a [[Finite field extension|finite]] [[Galois extension|Galois extension]] with [[Galois group of a field extension|Galois group]] $G=\operatorname{Gal}(L/K)$. 
Then, the map sending a [[Subgroup|subgroup]] $H$ to its associated [[Fixed field of a subgroup|fixed field]] $L^{H}$ is an order-reversing [[Bijective map|bijection]] between the set of subgroups of $G$ and the [[Intermediate field|intermediate fields]] $K \subseteq E\subseteq L$, whose inverse sends each intermediate field $E$ to its [[Fixing subgroup of an intermediate field|fixing subgroup]].  

##### Examples

##### Proof