Editing Central simple algebra (section)

{{Short description|Finite dimensional algebra over a field whose central elements are that field}}
In [[ring theory]] and related areas of [[mathematics]] a '''central simple algebra''' ('''CSA''') over a [[field (mathematics)|field]] ''K'' is a finite-dimensional [[associative algebra|associative ''K''-algebra]] ''A'' that is [[simple algebra|simple]], and for which the [[Center (ring theory)|center]] is exactly ''K''. (Note that ''not'' every simple algebra is a central simple algebra over its center: for instance, if ''K'' is a field of characteristic 0, then the [[Weyl algebra]] <math>K[X,\partial_X]</math> is a simple algebra with center ''K'', but is ''not'' a central simple algebra over ''K'' as it has infinite dimension as a ''K''-module.)

For example, the [[complex number]]s '''C''' form a CSA over themselves, but not over the [[real number]]s '''R''' (the center of '''C''' is all of '''C''', not just '''R'''). The [[quaternion]]s '''H''' form a 4-dimensional CSA over '''R''', and in fact represent the only non-trivial element of the [[Brauer group]] of the reals (see below).

Given two central simple algebras ''A'' ~ ''M''(''n'',''S'') and ''B'' ~ ''M''(''m'',''T'') over the same field ''F'', ''A'' and ''B'' are called ''similar'' (or ''[[Brauer equivalent]]'') if their division rings ''S'' and ''T'' are isomorphic. The set of all [[equivalence class]]es of central simple algebras over a given field ''F'', under this equivalence relation, can be equipped with a [[group operation]] given by the [[tensor product of algebras]]. The resulting group is called the [[Brauer group]] Br(''F'') of the field ''F''.<ref name=L159>Lorenz (2008) p.159</ref>  It is always a [[torsion group]].<ref name=L194>Lorenz (2008) p.194</ref>