Suppose that $D$ is a division algebra that is finite-dimensional over $\Bbb Q$, does there exist a finite group $G$ such that one of the factors in the Wedderburn decomposition of $\Bbb Q[G]$ is a matrix ring over $D$?

(Note that the answer with general base field is no, as there are countably many finite groups and each group algebra has only finitely factors in the Wedderburn decomposition, so the division algebras which appear as Wedderburn factors are countable, thus any field with uncountably many finite extensions is a counterexample)

The Schur subgroup of the Brauer group,LNM 397, Springer (1974), doi 10.1007/BFb0061703. $\endgroup$ – Frieder Ladisch May 31 '18 at 12:56