Let $F$ be a field, and let $A$ be a finite-dimensional central simple $F$-algebra. Then there exist an integer $r \in \mathbb N$ and a central division $F$-algebra $D$ such that
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\begin{align*}
A \cong M_r(D)
\end{align*}
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as $F$-algebras. Moreover, the division $F$-algebra $D$ is unique up to $F$-algebra isomorphism.