fundamental theorem of Galois theory
Let $L/F$ be a Galois extension^{} of finite degree, with Galois group^{} $G:=\mathrm{Gal}(L/F)$. There is a bijective^{}, inclusionreversing correspondence between subgroups^{} of $G$ and extensions^{} of $F$ contained in $L$, given by

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$K\mapsto \mathrm{Gal}(L/K)$, for any field $K$ with $F\subseteq K\subseteq L$.

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$H\mapsto {L}^{H}$ (the fixed field of $H$ in $L$), for any subgroup $H\le G$.
The extension ${L}^{H}/F$ is normal if and only if $H$ is a normal subgroup^{} of $G$, and in this case the homomorphism^{} $G\u27f6\mathrm{Gal}({L}^{H}/F)$ given by $\sigma \mapsto {\sigma }_{{L}^{H}}$ induces (via the first isomorphism theorem^{}) a natural identification $\mathrm{Gal}({L}^{H}/F)=G/H$ between the Galois group of ${L}^{H}/F$ and the quotient group^{} $G/H$.
For the case of Galois extensions of infinite^{} degree, see the entry on infinite Galois theory.
Title  fundamental theorem of Galois theory^{} 
Canonical name  FundamentalTheoremOfGaloisTheory 
Date of creation  20130322 12:08:31 
Last modified on  20130322 12:08:31 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  9 
Author  yark (2760) 
Entry type  Theorem 
Classification  msc 11S20 
Classification  msc 11R32 
Classification  msc 12F10 
Classification  msc 13B05 
Synonym  Galois theory 
Synonym  Galois correspondence 
Related topic  GaloisTheoreticDerivationOfTheCubicFormula 
Related topic  GaloisTheoreticDerivationOfTheQuarticFormula 
Related topic  InfiniteGaloisTheory 
Related topic  GaloisGroup 