Galois group
E308961
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Galois group canonical | 4 |
| Galois groups | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T2912347 — 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 group Context triple: [Abel–Ruffini theorem, usesConcept, Galois group]
-
A.
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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B.
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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C.
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.
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D.
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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E.
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.
- 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 group Target entity description: 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.
-
A.
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.
-
B.
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.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
field theory concept
ⓘ
group theory concept ⓘ mathematical concept ⓘ object in abstract algebra ⓘ |
| captures |
symmetries of field extensions
ⓘ
symmetries of roots of polynomials ⓘ |
| definition | group of field automorphisms of a field extension that fix the base field ⓘ |
| example |
Galois group of a finite field extension F_{q^n} over F_q is cyclic of order n
ⓘ
Galois group of x^2-2 over Q is of order 2 ⓘ absolute Galois group of a finite field is isomorphic to the profinite completion of Z ⓘ |
| field |
Galois theory
ⓘ
abstract algebra ⓘ algebraic geometry ⓘ algebraic number theory ⓘ number theory ⓘ |
| hasProperty |
can be infinite for infinite extensions
ⓘ
encodes solvability of polynomials by radicals ⓘ finite for finite Galois extensions ⓘ forms a group under composition of automorphisms ⓘ identity element is the identity automorphism ⓘ inverse of an element is its inverse automorphism ⓘ non-solvable Galois group can obstruct radical solutions ⓘ solvable Galois group implies solvability by radicals ⓘ |
| hasSubconcept |
absolute Galois group
ⓘ
decomposition group ⓘ geometric Galois group ⓘ inertia group ⓘ local Galois group ⓘ |
| namedAfter | Évariste Galois ⓘ |
| relatedTo |
Galois correspondence
ⓘ
Galois extension ⓘ absolute Galois group ⓘ automorphism ⓘ field extension ⓘ fundamental theorem of Galois theory ⓘ inverse limit ⓘ normal extension ⓘ normal subgroup ⓘ polynomial equation ⓘ profinite group ⓘ quotient group ⓘ separable extension ⓘ splitting field ⓘ |
| usedFor |
classifying field extensions
ⓘ
proving insolvability of general quintic by radicals ⓘ studying L-functions and representations ⓘ studying algebraic equations ⓘ studying algebraic number fields ⓘ studying ramification of primes ⓘ understanding fundamental groups in arithmetic geometry ⓘ |
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 group Description of subject: 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.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.