4
$\begingroup$

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)

$\endgroup$
9
$\begingroup$

The subgroup of the Brauer group generated by (and in fact, consisting of) such division algebras is called "the Schur group". Brauer-Witt theorem asserts that it is given by cyclotomic algebras, so the answer is negative. I learned this here.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.