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, inclusion-reversing correspondence between subgroups of $G$ and extensions of $F$ contained in $L$, given by

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

• $H\mapsto L^H$ (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 $G⟶\mathrm{Gal}(L^H/F)$ 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	2013-03-22 12:08:31
Last modified on	2013-03-22 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