Galois correspondence
E904574
Galois correspondence is a fundamental concept in field theory that establishes a one-to-one relationship between intermediate field extensions and subgroups of a Galois group.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Galois correspondence canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T11099162 — 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 correspondence Context triple: [Évariste Galois, conceptNamedAfter, Galois correspondence]
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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.
Galois cohomology
Galois cohomology is a branch of mathematics that studies Galois groups and their actions on modules using cohomological methods, providing powerful tools for understanding field extensions, algebraic number theory, and arithmetic geometry.
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D.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
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E.
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.
- 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 correspondence Target entity description: Galois correspondence is a fundamental concept in field theory that establishes a one-to-one relationship between intermediate field extensions and subgroups of a 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.
Galois cohomology
Galois cohomology is a branch of mathematics that studies Galois groups and their actions on modules using cohomological methods, providing powerful tools for understanding field extensions, algebraic number theory, and arithmetic geometry.
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D.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
-
E.
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in Galois theory
ⓘ
concept in field theory ⓘ mathematical concept ⓘ |
| appliesTo | Galois extensions ⓘ |
| assumes | field extensions are over a fixed base field ⓘ |
| centralTo |
inverse Galois problem
ⓘ
modern algebraic number theory ⓘ theory of finite fields ⓘ |
| characterizes | Galois extensions by the existence of a lattice isomorphism ⓘ |
| direction |
larger subgroups correspond to smaller intermediate fields
ⓘ
smaller subgroups correspond to larger intermediate fields ⓘ |
| domain | finite Galois extensions ⓘ |
| field |
Galois theory
NERFINISHED
ⓘ
abstract algebra ⓘ field theory ⓘ |
| formalizedAs | anti-isomorphism of lattices ⓘ |
| generalizedTo |
Galois theory of infinite extensions via Krull topology
ⓘ
infinite Galois extensions ⓘ |
| hasAnalogue | Galois connection in order theory ⓘ |
| hasVariant |
Galois correspondence for covering spaces in algebraic topology
ⓘ
Galois correspondence in algebraic geometry NERFINISHED ⓘ fundamental theorem of Galois theory ⓘ |
| historicalContext | developed in the 19th century ⓘ |
| implies |
normal subgroups correspond to normal intermediate extensions in towers
ⓘ
quotient groups correspond to subextensions ⓘ |
| involves |
Galois group
ⓘ
automorphisms of a field extension ⓘ fixed field of a subgroup ⓘ |
| is | a one-to-one correspondence between intermediate fields and subgroups of the Galois group ⓘ |
| maps |
each intermediate field to its stabilizer subgroup in the Galois group
ⓘ
each subgroup of the Galois group to its fixed field ⓘ |
| namedAfter | Évariste Galois NERFINISHED ⓘ |
| property |
inclusion-reversing
ⓘ
order-reversing ⓘ |
| relates |
intermediate fields
ⓘ
subgroups of a Galois group ⓘ |
| requires |
Galois group acts faithfully on the extension field
ⓘ
normality of the extension ⓘ separability of the extension ⓘ |
| structure | lattice anti-isomorphism between intermediate fields and subgroups ⓘ |
| typicalCondition |
the base field is contained in every intermediate field
ⓘ
the extension field is fixed by the trivial subgroup only ⓘ |
| typicalStatement | there is a bijection between intermediate fields of a finite Galois extension and subgroups of its Galois group ⓘ |
| usedIn |
classification of field extensions
ⓘ
construction of splitting fields ⓘ solvability of polynomial equations by radicals ⓘ study of normal closures ⓘ |
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 correspondence Description of subject: Galois correspondence is a fundamental concept in field theory that establishes a one-to-one relationship between intermediate field extensions and subgroups of a Galois group.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Évariste Galois