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- Hasse_norm_theorem abstract "In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm.Here to be a global norm means to be an element k of K such that there is an element l of L with in other words k is a relative norm of some element of the extension field L. To be a local norm means that for some prime p of K and some prime P of L lying over K, then k is a norm from LP; here the "prime" p can be an archimedean valuation, and the theorem is a statement about completions in all valuations, archimedean and non-archimedean.The theorem is no longer true in general if the extension is abelian but not cyclic. A counter-example is given by the field where every rational square is a local norm everywhere but is not a global norm.This is an example of a theorem stating a local-global principle, and is due to Helmut Hasse.The Hasse norm theorem can be deduced from the theorem that an element of the Galois cohomology group H2(L/K) is trivial if it is trivial locally everywhere, which is in turn equivalent to the deep theorem that the first cohomology of the idele class group vanishes. This is true for all finite Galois extensions of number fields, not just cyclic ones. For cyclic extensions the group H2(L/K) is isomorphic to the Tate cohomology group H0(L/K) which describes which elements are norms, so for cyclic extensions it becomes Hasse's theorem that an element is a norm if it is a local norm everywhere.".
- Hasse_norm_theorem wikiPageID "1108795".
- Hasse_norm_theorem wikiPageRevisionID "581719446".
- Hasse_norm_theorem hasPhotoCollection Hasse_norm_theorem.
- Hasse_norm_theorem subject Category:Class_field_theory.
- Hasse_norm_theorem subject Category:Theorems_in_algebraic_number_theory.
- Hasse_norm_theorem type Abstraction100002137.
- Hasse_norm_theorem type Communication100033020.
- Hasse_norm_theorem type Message106598915.
- Hasse_norm_theorem type Proposition106750804.
- Hasse_norm_theorem type Statement106722453.
- Hasse_norm_theorem type Theorem106752293.
- Hasse_norm_theorem type TheoremsInAlgebra.
- Hasse_norm_theorem type TheoremsInAlgebraicNumberTheory.
- Hasse_norm_theorem type TheoremsInNumberTheory.
- Hasse_norm_theorem comment "In number theory, the Hasse norm theorem states that if L/K is a cyclic extension of number fields, then if a nonzero element of K is a local norm everywhere, then it is a global norm.Here to be a global norm means to be an element k of K such that there is an element l of L with in other words k is a relative norm of some element of the extension field L.".
- Hasse_norm_theorem label "Hasse norm theorem".
- Hasse_norm_theorem label "Théorème de la norme de Hasse".
- Hasse_norm_theorem sameAs Théorème_de_la_norme_de_Hasse.
- Hasse_norm_theorem sameAs m.046r3g.
- Hasse_norm_theorem sameAs Q3527209.
- Hasse_norm_theorem sameAs Q3527209.
- Hasse_norm_theorem sameAs Hasse_norm_theorem.
- Hasse_norm_theorem wasDerivedFrom Hasse_norm_theorem?oldid=581719446.
- Hasse_norm_theorem isPrimaryTopicOf Hasse_norm_theorem.