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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Galois extension canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10055685 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Galois extension Context triple: [Die Theorie der algebraischen Zahlkörper, usesConcept, Galois extension]
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A.
Galois group
A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
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B.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
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C.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
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D.
Galois
Galois is a French surname most famously associated with Évariste Galois, the pioneering 19th-century mathematician who founded group theory and laid the groundwork for modern abstract algebra.
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E.
Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Galois extension Target entity description: 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.
-
A.
Galois group
A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
-
B.
Galois theory
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
-
C.
Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
-
D.
Galois
Galois is a French surname most famously associated with Évariste Galois, the pioneering 19th-century mathematician who founded group theory and laid the groundwork for modern abstract algebra.
-
E.
Kummer theory
Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Galois extension Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Évariste Galois