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In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

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rdf:type	Thing software
rdfs:label	Galois-Darstellung (de) Módulo de Galois (es) Galois module (en) Représentation galoisienne (fr) 갈루아 모듈 (ko) ガロワ加群 (ja) Galoismodul (sv)
rdfs:comment	Eine Galois-Darstellung ist eine Darstellung einer Galoisgruppe auf einem Vektorraum oder allgemeiner einem Modul über einem kommutativen Ring. Häufig fordert man Stetigkeit bezüglich der Krulltopologie der Galoisgruppe und einer Topologie auf dem Koeffizientenring. (de) In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. (en) La théorie des représentations galoisiennes est l'application naturelle de la théorie des représentations à la théorie algébrique des nombres. Un module galoisien est un module sur lequel agit un groupe de Galois G. Ces modules seront par exemple des groupes d'unités, des groupes des classes, ou des groupes de Galois eux-mêmes. (fr) 数学において、ガロワ加群 (Galois module) は、G がある体の拡大のガロワ群であるときの G-加群である。G-加群が体上のベクトル空間や環上の自由加群であるときに、用語ガロワ表現 (Galois representation) がしばしば用いられるが、G-加群の同義語としても用いられる。局所体や大域体の拡大のガロワ加群の研究は数論において重要なツールである。 (ja) 갈루아 모듈은 어떤 체의 체의 확대의 갈루아 군 G에 적용되는 군의 가군이다. (ko) Inom matematiken är en Galoismodul en där G är Galoisgruppen av någon kroppsutvidgning. Termen Galoisrepresentation används ofta när G-modulen är ett vektorrum över en kropp eller en fri modul över en ring, men används även som synonym till G-modul. Studien av Galoismoduler för utvidgningar av eller är ett viktigt område inom talteori. (sv) En matemáticas, y particularmente en la teoría de números algebraicos, un módulo de Galois es un módulo para un grupo de Galois G. En forma equivalente, para un grupo de Galois G y un A[G] de G con respecto a un cierto anillo A, un módulo de Galois es un cierto A[G] módulo M. En este sentido genérico, el término representación de Galois es un sinónimo. Existe un número importante de casos mucho más especializados. Ciertamente G puede o bien ser un grupo de Galois infinito profinito, o uno finito para una extensión de Galois finita L/K de campos. (es)
dcterms:subject	Algebraic number theory Galois theory
Wikipage page ID	486929 (xsd:integer)
Wikipage revision ID	1059090880 (xsd:integer)
Link from a Wikipage to another Wikipage	Representation theory Valued field Normal basis theorem David Hilbert Algebraic closure Ring of integers Vector space Elliptic curves Continuous function Mathematics Gaussian period Gaussian rational Semistable abelian variety Emil Artin Emmy Noether Endomorphism Modular form Nilpotent Roots of unity Compatible system of ℓ-adic representations Kronecker–Weber theorem Topological group G-module Galois cohomology Galois extension Galois group Irreducible representation Local class field theory Local field Abelianization Algebraic integer Algebraic number theory American Mathematical Society Cyclotomic field Field (mathematics) Finite extension Finitely generated module Brauer group Number field Number theory P-adic number Discriminant of an algebraic number field Global field Isomorphism Projective module Projective variety Proper morphism Ramification group Ring (mathematics) Group cohomology Group homomorphism Hilbert's theorem 90 Hilbert–Speiser theorem Isomorphic Prime number Abelian group Abelian variety Algebraic number theory Galois theory Absolute Galois group Holomorph (mathematics) Discrete topology Modular representation theory Artin L-function Artin conjecture (L-functions) Artin reciprocity law Class formation Field extension Free module Group scheme Integral closure Ramification (mathematics) Rational number Separable closure Separable extension Scheme (mathematics) Image (mathematics) Smooth morphism Unit group Tate module

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