Galois correspondence

E904574

concept in Galois theory concept in field theory mathematical concept

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.

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All labels observed (3)

Label	Occurrences
Galois correspondence canonical	2
Galois correspondence in algebraic geometry	1
fundamental theorem of Galois theory	1

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]

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.
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
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.
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 ⓘ 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 ⓘ linked to: Galois correspondence 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 ⓘ
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

Referenced by (4)

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

Évariste Galois → conceptNamedAfter → Galois correspondence ⓘ

subject linked to: Galois

Galois group → relatedTo → Galois correspondence ⓘ

Galois extension → satisfies → fundamental theorem of Galois theory ⓘ

linked to: Galois correspondence

Galois correspondence → hasVariant → Galois correspondence in algebraic geometry ⓘ