Galois theory | GPTKB 2.0

All labels observed (5)

Label	Occurrences
Galois theory canonical	21
Galois Theory	3
"Basic 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.
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
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

Referenced by (27)

Full triples — surface form annotated when it differs from this entity's canonical label.

construction of the regular 17-gon with straightedge and compass → isLinkedTo → Galois theory ⓘ

Évariste Galois → fieldOfWork → Galois theory ⓘ

Évariste Galois → notableWork → Galois theory ⓘ

Abel–Ruffini theorem → field → Galois theory ⓘ

Abel–Ruffini theorem → relatedTo → Galois’ theory of equations ⓘ

linked to: Galois theory

Emil Artin → fieldOfWork → Galois theory ⓘ

Ian Stewart → notableWork → Galois Theory ⓘ

linked to: Galois theory

Gaussian periods → relatedTo → Galois theory ⓘ

Galois theory → usesConcept → fundamental theorem of Galois theory ⓘ

linked to: Galois theory

quadratic reciprocity law → proofMethodsInclude → Galois theory ⓘ

quadratic reciprocity law → influencedDevelopmentOf → Galois theory ⓘ

Noether field → studiedIn → Galois theory ⓘ

Évariste Galois → knownFor → Galois theory ⓘ

subject linked to: Galois

Galois → hasNotableAssociation → Galois theory ⓘ

Galois group → field → Galois theory ⓘ

Artin–Schreier theory → relatesTo → Galois theory ⓘ

orbit-stabilizer theorem → usedIn → Galois theory ⓘ

A Survey of Modern Algebra → hasSubject → Galois theory ⓘ

Frobenius conjugacy class → definedInContextOf → Galois theory ⓘ

Nikolai Chebotaryov → notableWork → "Basic Galois Theory" ⓘ

linked to: Galois theory

Galois extension → studiedIn → Galois theory ⓘ

Harold M. Edwards → notableWork → Galois Theory ⓘ

linked to: Galois theory

Harold M. Edwards → hasWrittenOn → Galois theory ⓘ

H. M. Edwards → notableWork → Galois Theory ⓘ

linked to: Galois theory

reciprocity conjecture → field → Galois theory ⓘ

Galois correspondence → field → Galois theory ⓘ

Traité des substitutions et des équations algébriques → field → Galois theory ⓘ

All labels observed (5)

How this entity was disambiguated Show

Statements (49)

How these facts were elicited Show

Referenced by (27)

How this entity was disambiguated

How these facts were elicited