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单词 FundamentalTheoremOfGaloisTheory
释义

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 correspondencebetween 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|LHinduces (via the first isomorphism theoremPlanetmathPlanetmath)a natural identification Gal(LH/F)=G/Hbetween 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.

Titlefundamental theorem of Galois theoryMathworldPlanetmath
Canonical nameFundamentalTheoremOfGaloisTheory
Date of creation2013-03-22 12:08:31
Last modified on2013-03-22 12:08:31
Owneryark (2760)
Last modified byyark (2760)
Numerical id9
Authoryark (2760)
Entry typeTheorem
Classificationmsc 11S20
Classificationmsc 11R32
Classificationmsc 12F10
Classificationmsc 13B05
SynonymGalois theory
SynonymGalois correspondence
Related topicGaloisTheoreticDerivationOfTheCubicFormula
Related topicGaloisTheoreticDerivationOfTheQuarticFormula
Related topicInfiniteGaloisTheory
Related topicGaloisGroup
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