Difference between revisions of "Galois group"
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| − | The '''Galois group''' is a mathematical structure ([[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. | + | 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. |
Galois groups are named after [[Évariste Galois]], a [[French]] [[mathematician]]. | Galois groups are named after [[Évariste Galois]], a [[French]] [[mathematician]]. | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
Revision as of 21:08, November 17, 2008
The Galois group is a mathematical structure (group) associated to an extension of fields. 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 automorphisms 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.
Galois groups are named after Évariste Galois, a French mathematician.