Galois cohomology

E839567

branch of mathematics cohomology theory

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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Statements (60)

Predicate	Object
instanceOf	branch of mathematics ⓘ cohomology theory ⓘ
appliesTo	absolute Galois groups ⓘ algebraic extensions of fields ⓘ algebraic number fields ⓘ field extensions ⓘ global fields ⓘ local fields ⓘ
field	algebra ⓘ arithmetic geometry ⓘ number theory ⓘ
formalDefinition	right derived functors of the fixed-point functor for Galois modules ⓘ
hasCanonicalReference	Serre: Galois Cohomology NERFINISHED ⓘ
hasHistoricalDevelopmentBy	Chevalley NERFINISHED ⓘ Serre NERFINISHED ⓘ Tate NERFINISHED ⓘ
hasKeyConcept	Brauer group NERFINISHED ⓘ Galois module ⓘ Hilbert 90 NERFINISHED ⓘ Hochschild–Serre spectral sequence NERFINISHED ⓘ Kummer theory NERFINISHED ⓘ Poitou–Tate duality NERFINISHED ⓘ Shapiro lemma NERFINISHED ⓘ Tate cohomology NERFINISHED ⓘ Tate–Shafarevich group NERFINISHED ⓘ cohomological dimension of a field ⓘ cohomology group H^n(G,M) ⓘ continuous cochains ⓘ corestriction map ⓘ cup product ⓘ fundamental class in H^2 ⓘ inflation–restriction sequence ⓘ local Tate duality NERFINISHED ⓘ norm map ⓘ profinite Galois group ⓘ restriction map ⓘ
relatedTo	Galois representations NERFINISHED ⓘ K-theory of fields NERFINISHED ⓘ Selmer group NERFINISHED ⓘ Weil group NERFINISHED ⓘ class field theory ⓘ motivic cohomology ⓘ étale cohomology ⓘ
studies	Galois groups ⓘ actions of Galois groups on modules ⓘ continuous group cohomology of Galois groups ⓘ
typicalGroup	absolute Galois group of a field ⓘ
typicalModule	discrete Galois module GENERATED ⓘ finite Galois module GENERATED ⓘ p-adic Galois representation GENERATED ⓘ
usedIn	classification of central simple algebras ⓘ description of the Brauer group of a field ⓘ obstructions to local-global principles ⓘ study of principal homogeneous spaces ⓘ study of rational points on varieties ⓘ study of torsors under algebraic groups ⓘ
usesMethod	Ext functors ⓘ derived functors ⓘ group cohomology ⓘ homological algebra ⓘ

Referenced by (5)

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

Galois → conceptNamedAfter → Galois cohomology ⓘ

subject surface form: Évariste Galois

Hasse norm theorem → hasFormulationIn → Galois cohomology ⓘ

Iwasawa theory → relatedTo → Galois cohomology ⓘ

Milnor K-theory → relatedTo → Galois cohomology ⓘ

étale cohomology → relatedTo → Galois cohomology ⓘ