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)

Questions 2 and 3 are addressed in section 11 of

Auel, Asher; Brussel, Eric; Garibaldi, Skip; Vishne, Uzi, Open problems on central simple algebras., Transform. Groups 16, No. 1, 219-264 (2011). ZBL1230.16016.

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.

What I is very funny is that Ford has shown that every finite abelian group is the Brauer group of some commutative ring.

Edit: At the end of section 11 there is a reference to the very conveniently titled

Efrat, Ido, On fields with finite Brauer groups, Pac. J. Math. 177, No.1, 33-46 (1997). ZBL0868.12005.

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