If $F$ is a non-Archimedean local field, then any inner form of $GL\_n(F)$ is isomorphic to $GL\_m(D)$, for a central $F$-division algebra of dimension $d^2, md=n$. Why is this true? I looked in Platonov-Rapinchuk and couldn't find the answer.



Inner forms of $GL_n(F)$
This is Proposition 2.17 on page 87 of Platonov-Rapinchuk (for $SL\_n(F)$, instead of $GL\_n(F)$). There is no need to restrict to local fields.


