Galois theory
E157385
Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Galois theory canonical | 12 |
| Galois Theory | 1 |
| Galois’ theory of equations | 1 |
| fundamental theorem of Galois theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1382384 — 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 theory Context triple: [construction of the regular 17-gon with straightedge and compass, isLinkedTo, Galois theory]
-
A.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
B.
Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
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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.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- 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 theory Target entity description: Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
-
A.
Noether's problem
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
B.
Abel–Ruffini theorem
The Abel–Ruffini theorem is a fundamental result in algebra proving that there is no general solution in radicals for polynomial equations of degree five or higher.
-
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.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
Erlangen Program
The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
branch of abstract algebra
ⓘ
branch of mathematics ⓘ |
| addresses | solvability of polynomial equations by radicals ⓘ |
| appliesTo |
algebraic extensions
ⓘ
algebraic number fields ⓘ finite fields ⓘ function fields ⓘ |
| centralConcept |
Galois correspondence
ⓘ
Galois group ⓘ automorphism ⓘ discriminant ⓘ field extension ⓘ fixed field ⓘ intermediate field ⓘ normal extension ⓘ resolvent ⓘ separable extension ⓘ splitting field ⓘ |
| characterizedBy |
Galois correspondence between subgroups and intermediate fields
ⓘ
correspondence between normal subgroups and subextensions ⓘ |
| field | abstract algebra ⓘ |
| hasApplication |
classification of finite extensions of the rationals
ⓘ
construction of ruler-and-compass constructible numbers ⓘ |
| hasSubfield |
algebraic number theory applications
ⓘ
classical Galois theory ⓘ differential Galois theory ⓘ infinite Galois theory ⓘ inverse Galois theory ⓘ |
| historicalOrigin | 19th century ⓘ |
| implies | general quintic is not solvable by radicals ⓘ |
| namedAfter | Évariste Galois ⓘ |
| relatedTo |
field automorphisms
ⓘ
group actions ⓘ solvable groups ⓘ |
| relates | field extensions to groups of automorphisms ⓘ |
| studies |
field extensions
ⓘ
polynomial equations ⓘ symmetry of roots of polynomials ⓘ |
| toolFor |
algebraic geometry
ⓘ
algebraic number theory ⓘ coding theory ⓘ cryptography ⓘ |
| usesConcept |
field theory
ⓘ
fundamental theorem of Galois theory ⓘ
linked to:
Galois theory
group theory ⓘ normal closure ⓘ primitive element theorem ⓘ ring theory ⓘ separability ⓘ |
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 theory Description of subject: Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
Referenced by (15)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Galois theory
linked to: Galois theory
linked to: Galois theory