# fundamental theorem of Galois theory

Let $L/F$ be a Galois extension of finite degree, with Galois group $G:=\operatorname{Gal}(L/F)$. There is a bijective, inclusion-reversing correspondence between subgroups of $G$ and extensions of $F$ contained in $L$, given by

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

• $H\mapsto L^{H}$ (the fixed field of $H$ in $L$), for any subgroup $H\leq G$.

The extension $L^{H}/F$ is normal if and only if $H$ is a normal subgroup of $G$, and in this case the homomorphism $G\longrightarrow\operatorname{Gal}(L^{H}/F)$ given by $\sigma\mapsto\sigma|_{L^{H}}$ induces (via the first isomorphism theorem) a natural identification $\operatorname{Gal}(L^{H}/F)=G/H$ 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