Galois correspondence theorem

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7/7/2026, 12:47:03 PM · Ancient Tree

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Let $K \subseteq L$ a [[Finite field extension|finite]] [[Galois extension|Galois extension]] with [[Galois group|Galois group]] $G=\operatorname{Gal}(L/K)$.

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]].

4##### Examples

6##### Proof

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7/7/2026, 12:35:53 PM · Ancient Tree

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Let $K \subseteq L$ a [[Finite field extension|finite]] [[Galois extension|Galois extension]] with [[Galois group|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