This question is just a curiosity, but I'm really interested in the answer. It was originally posted on math.stackexchange (<https://math.stackexchange.com/questions/368897/inverse-problem-for-brauer-groups>), but hasn't received any responses despite some upvotes, so I'm posting it here.


Given a field $K$, we can form the set$^\*$ $Br(K)$ consisting of equivalence classes of finite-dimensional central simple $K$-algebras which split over some Galois extension of $K$, modulo "are Morita-equivalent" (I hope I have that right, it's been a while). This set is actually a group, in a natural way: the tensor product over $K$ is well-defined on the equivalence classes, and has identity (the equivalence class of $K$ as an algebra over itself) and inverses (given by $R\mapsto R^{op}$). Actually $Br(K)$ turns out to be a second cohomology group, in a natural and useful way, but I don't really have a good understanding of that part.


My main question is, what groups are the Brauer group of some field? I know a couple trivial bits of the answer to this: $Br(K)$ is always abelian, and of cardinality at most $\aleph\_0\times\vert K\vert$, and $Br(K)$ is always torsion. Within those constraints, I only know of one specific nontrivial Brauer group: $Br(\mathbb{R})=\mathbb{Z}/2\mathbb{Z}$ by Frobenius' Theorem on division algebras over $\mathbb{R}$. (I've seen the Brauer group of $\mathbb{Q}$ described by a short exact sequence, but I wasn't able to get an explicit description from that; is it known?) EDIT: As Emerton points out in a comment below, the Brauer group of $\mathbb{Q}$ (and much more) is known: it is $Br(\mathbb{Q})=\mathbb{Z}/2\mathbb{Z}\oplus\bigoplus\_1^\infty\mathbb{Q}/\mathbb{Z}.$


My main question is: is there a known list of properties which are necessary and sufficient for a group to be $\cong Br(K)$ for some $K$?


There are many possible variations/elaborations of this question, which may not have deep significance but seem kind of interesting. For example, leaving the context of fields for a moment, there is an analogous notion of Brauer group for groups, and we can ask (although I'm not sure *why* we would ask): is there a group which is its own Brauer group? My second question is just: is there a good resource for this type of question, that is, for constructing Brauer groups of various objects to specification? I imagine the opposite direction (finding Brauer groups of fields we already care about) is much more useful, but I'm personally interested in this direction.


(As an aside, I'm not sure whether the "group theory" tag is appropriate here; if it is not, feel free to delete it, or let me know and I will delete it.)




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$^\*$ As a very minor aside, note that size issues don't arise here: since we specify "finite-dimensional," there are at most $\aleph\_0\times\vert K\vert$ many such algebras up to isomorphism; using Scott's trick then lets us represent these equivalence classes in perfectly fine way.



"Inverse problem" for Brauer groups
It is perhaps helpful to look at the following papers of Fein and Schacher:



> 
> 
> > 
> > Fein; Schacher;
> >  Brauer groups of fields algebraic over Q.
> >  J. Algebra 43 (1976), no. 1, 328–337.
> > 
> > 
> > Fein; Schacher;
> >  Divisible groups that are Brauer groups.
> >  Commun. Algebra 7, 989-994 (1979).
> > 
> > 
> > 
> 
> 
> 


I doesn't have the first paper but according to MathSciNet review of the first item an abelian countable group $G$ is the Brauer group of some field $K$ if and only if $G$ is the direct sum of a divisible group and a $2$-torsion group. Moreover $K$ can be chosen to be an algebraic extension of rational numbers (this is mentioned in the second paper).


Also the authors of the following paper   




> 
> 
> > 
> > Auel; Brussel; Garibaldi; Vishne;
> >  Open problems on central simple algebras.
> >  Transform. Groups 16 (2011), no. 1, 219–264.
> >    
> > 
> > 
> > 
> > 
> 
> 
> 


cite a Russian paper of Merkurjev (published in 1985) in which it has been proved that every divisible torsion abelian group is the Brauer group of certain field.


