# Some questions on division algebras

1. Given a field $K$, is there a finite dimensional quiver algebra, such that any finite dimensional division algebra is isomorphic to End(M)/rad(End(M)) for some indecomposable finite dimensional module M?
2. Can fields with finite Brauer group be somehow characterised?
3. What are the field with Brauer group equal to $\mathbb{Z}/\mathbb{Z}3$? I do not even know one example. ($\mathbb{Z}/\mathbb{Z}3$ might be replaced by any finite abelian group with at least 3 elements)

In particular, the Brumer-Rosen conjecture implies that there are no fields with Brauer group $\mathbb{Z}/3\mathbb{Z}$ (and this is known), and only elementary abelian 2-groups appear. But this is still a conjecture, and $\mathbb{Z}/7\mathbb{Z}$ isn't ruled out yet.