Editing Central simple algebra (section)

==Splitting field==
We call a field ''E'' a ''splitting field'' for ''A'' over ''K'' if ''A''⊗''E'' is isomorphic to a matrix ring over ''E''.  Every finite dimensional CSA has a splitting field: indeed, in the case when ''A'' is a division algebra, then a [[maximal subfield]] of ''A'' is a splitting field.  In general by theorems of [[Joseph Wedderburn|Wedderburn]] and Koethe there is a splitting field which is a [[separable extension]] of ''K'' of degree equal to the index of ''A'', and this splitting field is isomorphic to a subfield of ''A''.<ref name=Jac2728>Jacobson (1996) pp.27-28</ref><ref name=GS101>Gille & Szamuely (2006) p.101</ref>  As an example, the field '''C''' splits the quaternion algebra '''H''' over '''R''' with

:<math> t + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \leftrightarrow
\left({\begin{array}{*{20}c} t + x i & y + z i \\ -y + z i & t - x i \end{array}}\right) . </math>

We can use the existence of the splitting field to define '''reduced norm''' and '''reduced trace''' for a CSA ''A''.<ref name=GS378>Gille & Szamuely (2006) pp.37-38</ref>  Map ''A'' to a matrix ring over a splitting field and define the reduced norm and trace to be the composite of this map with determinant and trace respectively.  For example, in the quaternion algebra '''H''', the splitting above shows that the element ''t'' + ''x'' '''i''' + ''y'' '''j''' + ''z'' '''k''' has reduced norm ''t''<sup>2</sup> + ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> and reduced trace 2''t''.

The reduced norm is multiplicative and the reduced trace is additive.  An element ''a'' of ''A'' is invertible if and only if its reduced norm in non-zero: hence a CSA is a division algebra if and only if the reduced norm is non-zero on the non-zero elements.<ref name=GS38>Gille & Szamuely (2006) p.38</ref>