How does Wedderburns's little Theorem imply that central simple Algebras are isomorphic to matrix Algebras over a divison Algebra?

Question

For a thesis I need the statement that every central simple algebra $A$ is isomorphic to $M_n(D)$ for some divison algebra $D$ and integer $n$. In a reference I am reading it states that this is a corollary from Wedderburn's little theorem i.e. every finite divion ring is a field. But there is no proof given which should mean it is quite easy to see. But I just can't figure it out!

Answer 1

This is a corollary of a different Wedderburn theorem (sometimes called the Artin-Wedderburn theorem or the Wedderburn decomposition theorem), not the one about finite division rings. The correct statement is

A finite-dimensional $K$-algebra $A$ is a simple if and only if $A\cong M_n(D)$ for a finite-dimensional division $K$-algebra $D$. The number $n$ and the isomorphism class of $D$ are uniquely determined and $A$ is central if and only if $D$ is.

For a proof of this see Falko Lorenz's "Algebra II" (freely available here), Theorem $5$ on page $158$.