# Central simple algebra

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2010 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL]

A central simple algebra is a simple associative algebra with a unit element (cf. Simple algebra, that is a central algebra. Every finite-dimensional central simple algebra $A$ over a field $K$ is isomorphic to a matrix algebra $M_n(C)$ over a finite-dimensional central division algebra $C$ over $K$. In particular, if $K$ is algebraically closed, then every finite-dimensional central simple algebra $A$ over $K$ is isomorphic to $M_n(K)$, and if $K=\R$, then $A$ is isomorphic to the algebra of real or quaternion matrices. The tensor product of a central simple algebra $A$ and an arbitrary simple algebra $B$ is a simple algebra, which is central if $B$ is central. Two finite-dimensional central simple algebras $A$ and $B$ over $K$ are called equivalent if $$A\otimes_K M_m(K) \cong B\otimes M_n(K)$$ for certain $m$ and $n$, or, which is equivalent, if $A$ and $B$ are isomorphic matrix algebras over one and the same central division algebra. The equivalence classes of central simple algebras over $K$ form the Brauer group of $K$ relative to the operation induced by tensor multiplication.