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In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor.

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rdf:type	yago:WikicatCohomologyTheories yago:Abstraction100002137 yago:Cognition100023271 yago:Explanation105793000 yago:HigherCognitiveProcess105770664 yago:Process105701363 yago:PsychologicalFeature100023100 book yago:Theory105989479 yago:Thinking105770926
rdfs:label	Galoiskohomologie (de) Cohomologie galoisienne (fr) Galois cohomology (en) ガロワコホモロジー (ja) Galoiskohomologi (sv) 伽羅瓦上同調 (zh)
rdfs:comment	Unter Galoiskohomologie versteht man im mathematischen Teilgebiet der Zahlentheorie das Studium der Gruppenkohomologie von Galoisgruppen. Ist L\|K eine Körpererweiterung und A ein Galoismodul, also ein Modul unter der Galoisgruppe Gal(L\|K), so schreibt man (zur Notation siehe den Artikel Gruppenkohomologie) Ist speziell L = Ksep ein separabler Abschluss von K, so schreibt man auch Eines der ersten Resultate der Galoiskohomologie ist Hilberts Satz 90, der besagt: . Vor allem in der Klassenkörpertheorie ist die Beziehung zwischen Galoiskohomologie und Brauergruppe wichtig: . (de) In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group G associated to a field extension L/K acts in a natural way on some abelian groups, for example those constructed directly from L, but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. (en) 数学において、ガロワコホモロジー (Galois cohomology) はガロワ加群の群コホモロジーの研究、つまり、ホモロジー代数学のガロワ群に対する加群への応用である。体拡大 L/K と結びついたガロワ群 G はあるアーベル群、例えば L から直接に構成されたアーベル群、に自然に作用するが、より抽象的な手段によって導き出される他のガロワ表現を通して構成されたアーベル群もである。ガロワコホモロジーはガロワ不変元をとることが完全関手でなくなる理由を説明する。 (ja) Inom matematiken är Galoiskohomologi studien av av Galoismoduler, det vill säga användning av homologisk algebra till moduler för Galoisgrupper. (sv) 在數學中，伽羅瓦上同調是一套用群上同調研究伽羅瓦群的作用的技術。具體言之，假設伽羅瓦群作用在一個群（通常是數論中出現的代數結構，如等等）上，伽羅瓦上同調研究相關的群上同調。這些群通常具有重要的數論或算術代數幾何意義。伽羅瓦上同調是現代代數數論的基石之一。 (zh) En mathématiques, la cohomologie galoisienne est l'étude de l'action d'un groupe de Galois sur certains groupes, par des méthodes cohomologiques. Elle permet d'obtenir des résultats à la fois sur le groupe de Galois agissant, et sur le groupe sur lequel il agit. En particulier, le groupe de Galois d'une extension de corps de nombres L/K agit naturellement par exemple sur le groupe multiplicatif L×, mais aussi sur le groupe des unités de l'anneau des entiers du corps L, ou sur son groupe des classes. (fr)
dct:subject	Cohomology theories Homological algebra Algebraic number theory Class field theory Galois theory
Wikipage page ID	1111315 (xsd:integer)
Wikipage revision ID	1081674565 (xsd:integer)
Link from a Wikipage to another Wikipage	Module (mathematics) Multiplicative group Mordell–Weil theorem Normal basis theorem Birch and Swinnerton-Dyer conjecture Algebraic closure Algebraic group John Tate (mathematician) Richard Dedekind L-function Profinite group Mathematics Non-abelian class field theory Nonabelian cohomology Cohomology theories Elliptic curve Emil Artin Emmy Noether Étale cohomology Ideal class group Kummer theory Torsor Galois group Galois module Algebraic number theory Fermat Brauer group Hilbert's Theorem 90 Infinite descent Quadratic form Group cohomology Isomorphic Tate–Shafarevich group Homological algebra Abelian group Algebraic number theory Class field theory Galois theory Karl Rubin Homological algebra Class field theory Class formation Field extension Algebraically closed Selmer group Simple algebra Local-global principle Exact functor Patrick Ion Severi–Brauer variety Springer-Verlag Galois representation Hasse's norm theorem Arithmetic of elliptic curves Langlands philosophy Tate–Poitou duality Additive group of a ring dbr:Power_map
Link from a Wikipage to an external page	http://www.jmilne.org/math/Books/adt.html
sameAs	Galois cohomology Galois cohomology Galois cohomology Galois cohomology Galois cohomology Galois cohomology Galois cohomology Galois cohomology Galois cohomology
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Neukirch_et_al._CNF dbt:Numtheory-stub

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