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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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 ⓘ

Referenced by (2)

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

Galois → conceptNamedAfter → Galois correspondence ⓘ

subject surface form: Évariste Galois

Galois group → relatedTo → Galois correspondence ⓘ