Realising finite abelian groups as Brauer groups of a field

Question

Which finite abelian groups appear as Brauer groups of a field? Given a finite abelian group $G$, what are the (easiest) examples of fields with Brauer group equal to $G$?

Answer 1

There are two easy examples to see $Br(\Bbb{C})=\{1\}$ and $Br(\Bbb{R})=\Bbb{Z}/2\Bbb{Z}.$

For any nonarchimedian local field, $ \Bbb{F},$ $Br(\Bbb{F})\cong \Bbb{Q}/\Bbb{Z},$ but this group is infinite.

For any finite field $\Bbb{F},$ $Br(\Bbb{F})\cong \{1\},$ since Webberburn's theorem forces all finite division algebras to be fields.

I can't say I know of any Brauer groups which are finite groups other than the ones I've shown here.

You should consult http://stacks.math.columbia.edu/download/brauer.pdf if you'd like to see more about these facts, as well as http://link.springer.com/chapter/10.1007%2F978-0-387-72488-1_12#page-1

I'm not an expert on the Brauer group, but my intuition is that there aren't a wealth of examples, since the construction of the Brauer group relies on Galois Cohomology, in particular the second cohomology of the separable closure of the field in question . . .