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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.

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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.

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