Galois extension
E838591
A Galois extension is a field extension that is both normal and separable, characterized by a well-structured group of automorphisms known as its Galois group.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
field theory concept
ⓘ
mathematical notion ⓘ |
| characterizedBy | Galois group ⓘ |
| correspondsTo |
lattice of intermediate fields
ⓘ
lattice of subgroups of the Galois group ⓘ |
| definedAs | a field extension that is both normal and separable ⓘ |
| equivalentCondition |
the fixed field of the Galois group equals the base field
ⓘ
the number of K-embeddings of the extension into an algebraic closure equals the degree of the extension ⓘ |
| fieldOfStudy |
abstract algebra
ⓘ
field theory ⓘ |
| generalizes | splitting field of a separable polynomial ⓘ |
| hasAssociatedGroup | Galois group ⓘ |
| hasAssociatedStructure | group of field automorphisms ⓘ |
| hasBaseField | ground field ⓘ |
| hasBijectionWith | subgroups of its Galois group ⓘ |
| hasCondition |
every irreducible polynomial over the base field that has a root in the extension splits completely in the extension
ⓘ
the extension is separable over the base field ⓘ |
| hasConstraint | base field often assumed perfect in many classical treatments ⓘ |
| hasDegreeEqualTo | order of the Galois group for finite extensions ⓘ |
| hasExample |
cyclotomic field extension
ⓘ
extension of the rationals by adjoining all roots of a separable polynomial ⓘ finite extension of finite fields ⓘ quadratic extension with discriminant not equal to zero in characteristic not equal to 2 ⓘ |
| hasExtensionField | larger field ⓘ |
| hasGaloisGroup | finite group when the extension is finite ⓘ |
| hasIntermediateFieldsCorrespondingTo | subgroups of the Galois group ⓘ |
| hasInvariant | Galois group up to isomorphism ⓘ |
| hasNormalSubextensionsCorrespondingTo | normal subgroups of the Galois group ⓘ |
| hasProperty |
algebraic extension
ⓘ
normal extension ⓘ normal over the base field ⓘ separable extension ⓘ separable over the base field ⓘ |
| hasTypicalNotation | L over K with L∕K Galois ⓘ |
| implies | the extension is algebraic ⓘ |
| namedAfter | Évariste Galois NERFINISHED ⓘ |
| relatedTo |
automorphism group of a field
ⓘ
normal extension ⓘ separable extension ⓘ splitting field ⓘ |
| requires |
closure under all embeddings into an algebraic closure
ⓘ
separability of minimal polynomials ⓘ |
| satisfies | fundamental theorem of Galois theory NERFINISHED ⓘ |
| studiedIn | Galois theory NERFINISHED ⓘ |
| usedIn |
algebraic geometry
ⓘ
algebraic number theory ⓘ classification of field extensions ⓘ solvability of polynomial equations by radicals ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Évariste Galois