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| 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]] ''A'', which is [[simple algebra|simple]], and for which the [[center of an algebra|center]] is exactly ''K''. In other words, any simple algebra is a central simple algebra over its center.
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| 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).
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| 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>
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| == 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>
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| * Every [[automorphism]] of a central simple algebra is an [[inner automorphism]] (follows from [[Skolem–Noether theorem]]).
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| * The 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. 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>
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| * The ''period'' 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, and the two numbers are composed of the same prime factors.<ref name=GS104>Gille & Szamuely (2006) p.104</ref>
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| * If ''S'' is a simple [[subalgebra]] of a central simple algebra ''A'' then dim<sub>''F''</sub> ''S'' divides dim<sub>''F''</sub> ''A''.
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| * 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]].
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| * If ''D'' is a central division algebra over ''K'' for which the index has prime factorisation
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| ::<math>\mathrm{ind}(D) = \prod_{i=1}^r p_i^{m_i} \ </math>
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| :then ''D'' has a tensor product decomposition
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| ::<math>D = \otimes_{i=1}^r D_i \ </math>
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| :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>
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| ==Splitting field==
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| 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 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=GS101>Gille & Szamuely (2006) p.101</ref> As an example, the field '''C''' splits the quaternion algebra '''H''' over '''R''' with
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| :<math> t + x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \leftrightarrow
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| \left({\begin{array}{*{20}c} t + x i & y + z i \\ -y + z i & t - x i \end{array}}\right) . </math>
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| 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''.
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| 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>
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| == Generalization ==
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| CSAs over a field ''K'' are a non-commutative analog to [[extension field]]s over ''K'' – in both cases, they have no non-trivial 2-sided ideals, and have a distinguished field in their center, though a CSA can be non-commutative and need not have inverses (need not be a [[division algebra]]). This is of particular interest in [[noncommutative number theory]] as generalizations of [[number field]]s (extensions of the rationals '''Q'''); see [[noncommutative number field]].
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| == See also ==
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| * [[Azumaya algebra]], generalization of CSAs where the base field is replaced by a commutative local ring
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| * [[Severi–Brauer variety]]
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| ==References==
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| {{reflist}}
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| * {{cite book | title=Further Algebra and Applications | first=P.M. | last=Cohn | authorlink=Paul Cohn | edition=2nd | publisher=Springer | year=2003 | isbn=1852336676 | zbl=1006.00001 }}
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| * {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | publisher=American Mathematical Society | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
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| * {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | zbl=1130.12001 }}
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| ===Further reading===
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| * {{cite book | title=Structure of Algebras | volume=24 | series=Colloquium Publications | first=A.A. | last=Albert | authorlink=Abraham Adrian Albert | edition=7th revised reprint | publisher=American Mathematical Society | year=1939 | isbn=0-8218-1024-3 | zbl=0023.19901 }}
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| * {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
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| [[Category:Algebras]]
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| [[Category:Ring theory]]
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