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

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.