# Do all finite-dimensional division algebras appear as Wedderburn factors of rational group rings?

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 results of the paper ac.els-cdn.com/0022314X73900437/… would seem to indicate that not all appear. May 31 '18 at 2:20
• @BenjaminSteinberg this answers the question, thanks! May 31 '18 at 2:23
• A standard reference for this topic is Yamada, The Schur subgroup of the Brauer group, LNM 397, Springer (1974), doi 10.1007/BFb0061703. May 31 '18 at 12:56