Editing Central simple algebra (section)

==Properties==
* According to the [[Artin–Wedderburn theorem]] a finite-dimensional simple algebra ''A'' is isomorphic to the matrix algebra [[matrix ring|''M''(''n'',''S'')]] for some [[division ring]] ''S''. Hence, there is a unique division algebra in each Brauer equivalence class.<ref name=L160>Lorenz (2008) p.160</ref>
* Every [[automorphism]] of a central simple algebra is an [[inner automorphism]] (this follows from the [[Skolem–Noether theorem]]).
* The [[Dimension (vector space)|dimension]] of a central simple algebra as a vector space over its centre is always a square: the '''degree''' is the square root of this dimension.<ref name=GS21>Gille & Szamuely (2006) p.21</ref>  The '''Schur index''' of a central simple algebra is the degree of the equivalent division algebra:<ref name=L163>Lorenz (2008) p.163</ref> it depends only on the [[Brauer class]] of the algebra.<ref name=GS100>Gille & Szamuely (2006) p.100</ref>
* The '''period''' or '''exponent''' of a central simple algebra is the order of its Brauer class as an element of the Brauer group.  It is a divisor of the index,<ref name=Jac60>Jacobson (1996) p.60</ref> and the two numbers are composed of the same prime factors.<ref name=Jac61>Jacobson (1996) p.61</ref><ref name=GS104>Gille & Szamuely (2006) p.104</ref><ref>{{cite book | title=Further Algebra and Applications | first=Paul M. | last=Cohn | publisher=[[Springer-Verlag]] | year=2003 | isbn=1852336676 | page=208 |url=https://books.google.com/books?id=2Z_OC6uGzkwC&q=%22central+simple%22}}</ref>
* If ''S'' is a simple [[subalgebra]] of a central simple algebra ''A'' then dim<sub>''F''</sub>&nbsp;''S'' divides dim<sub>''F''</sub>&nbsp;''A''.
* Every 4-dimensional central simple algebra over a field ''F'' is isomorphic to a [[quaternion algebra]]; in fact, it is either a two-by-two [[matrix algebra]], or a [[division algebra]].
* If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation
::<math>\mathrm{ind}(D) = \prod_{i=1}^r p_i^{m_i} \ </math>
:then ''D'' has a tensor product decomposition
::<math>D = \bigotimes_{i=1}^r D_i \ </math>
:where each component ''D''<sub>''i''</sub> is a central division algebra of index <math>p_i^{m_i}</math>, and the components are uniquely determined up to isomorphism.<ref name=GS105>Gille & Szamuely (2006) p.105</ref>