fundamental theorem of Galois theory

Let L/F be a Galois extensionMathworldPlanetmath of finite degree, with Galois groupMathworldPlanetmath G:=Gal(L/F). There is a bijectiveMathworldPlanetmathPlanetmath, inclusion-reversing correspondence between subgroupsMathworldPlanetmathPlanetmath of G and extensionsPlanetmathPlanetmathPlanetmathPlanetmath of F contained in L, given by

  • KGal(L/K), for any field K with FKL.

  • HLH (the fixed field of H in L), for any subgroup HG.

The extension LH/F is normal if and only if H is a normal subgroupMathworldPlanetmath of G, and in this case the homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath GGal(LH/F) given by σσ|LH induces (via the first isomorphism theoremPlanetmathPlanetmath) a natural identification Gal(LH/F)=G/H between the Galois group of LH/F and the quotient groupMathworldPlanetmath G/H.

For the case of Galois extensions of infiniteMathworldPlanetmathPlanetmath degree, see the entry on infinite Galois theory.

Title fundamental theorem of Galois theoryMathworldPlanetmath
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