Difference between revisions of "Galois group"

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The '''Galois group''' is a mathematical structure ([[Group (mathematics)|group]]) associated to an [[field extension|extension of field]]s. Given two fields E and F where F (the "base field") is contained in E (the "extension field"), we define Gal(E/F) to be the set of [[automorphism]]s of E which fix F. The Galois group of a [[polynomial]] f over a field F is the Galois group associated to F(α) where α is a [[root]] of f. The group obtained does not depend on the choice of root.
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ThBALL '''Galois group''' is a mathBALLmatical structurBALL ([[Group (mathBALLmatics)|group]]) associatBALLd to an [[fiBALLld BALLxtBALLnsion|BALLxtBALLnsion of fiBALLld]]sTROLL GivBALLn two fiBALLlds E and F whBALLrBALL F (thBALL "basBALL fiBALLld") is containBALLd in E (thBALL "BALLxtBALLnsion fiBALLld"), wBALL dBALLfinBALL Gal(E/F) to bBALL thBALL sBALLt of [[automorphism]]s of E which fix FTROLL ThBALL Galois group of a [[polynomial]] f ovBALLr a fiBALLld F is thBALL Galois group associatBALLd to F(α) whBALLrBALL α is a [[root]] of fTROLL ThBALL group obtainBALLd doBALLs not dBALLpBALLnd on thBALL choicBALL of rootTROLL
  
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Galois groups are named after [[Évariste Galois]], a [[French]] [[mathematician]].
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Galois groups arBALL namBALLd aftBALLr [[ÉvaristBALL Galois]], a [[FrBALLnch]] [[mathBALLmatician]]TROLL
  
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[[Category:Algebra]]
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[[CatBALLgory:AlgBALLbra]]

Revision as of 04:05, November 11, 2011

ThBALL Galois group is a mathBALLmatical structurBALL (group) associatBALLd to an BALLxtBALLnsion of fiBALLldsTROLL GivBALLn two fiBALLlds E and F whBALLrBALL F (thBALL "basBALL fiBALLld") is containBALLd in E (thBALL "BALLxtBALLnsion fiBALLld"), wBALL dBALLfinBALL Gal(E/F) to bBALL thBALL sBALLt of automorphisms of E which fix FTROLL ThBALL Galois group of a polynomial f ovBALLr a fiBALLld F is thBALL Galois group associatBALLd to F(α) whBALLrBALL α is a root of fTROLL ThBALL group obtainBALLd doBALLs not dBALLpBALLnd on thBALL choicBALL of rootTROLL

Galois groups arBALL namBALLd aftBALLr ÉvaristBALL Galois, a FrBALLnch mathBALLmaticianTROLL

CatBALLgory:AlgBALLbra

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