# Brauer Group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer. The group may also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.

### Other articles related to "brauer group, group, brauer groups, groups":

Azumaya Algebra
... made it the basis for his geometric theory of the Brauer group in Bourbaki seminars from 1964-5 ... The Brauer group of X (an analogue of the Brauer group of a field) is the set of equivalence classes of Azumaya algebras ... The group operation is given by tensor product, and the inverse is given by the opposite algebra ...
Class Formation - The Brauer Group
... The Brauer groups H2(E/*) of a class formation are defined to be the direct limit of the groups H2(E/F) as F runs over all open subgroups of E ... An easy consequence of the vanishing of H1 for all layers is that the groups H2(E/F) are all subgroups of the Brauer group ... In local class field theory the Brauer groups are the same as Brauer groups of fields, but in global class field theory the Brauer group of the formation is not the Brauer group of the ...
Brauer Group - General Theory
... For an arbitrary field K, the Brauer group may be expressed in terms of Galois cohomology as follows Here, Ks is the separable closure of K, which coincides with the algebraic closure ... A generalisation of the Brauer group to the case of commutative rings by M ...

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