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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.)

$^*$ 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.

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    $\begingroup$ Dear Noah, The Brauer groups of all local and global fields (e.g. all number fields, including $\mathbb Q$) are known, by local/global class field theory. Regards, $\endgroup$
    – Emerton
    Commented May 16, 2013 at 2:04
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    $\begingroup$ B(Q) is not quite equal to the direct sum you write in the edit in response to Emerton's comment. It injects into the direct sum, and the image is equal to the kernel of the "sum of the components" from this direct sum onto Q/Z. This is a way of expressing the fact that a central simple algebra over Q gives rise to a family of central simple algebras over Q_p (for every prime p, including p=\infty), but you don't get all such local families from a global object, $\endgroup$
    – Chandan Singh Dalawat
    Commented May 16, 2013 at 3:05
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    $\begingroup$ B(Q) might be abstractly isomorphic to this direct sum (and that might suffice for an algebraist), but the point is that B(Q_p) is naturally isomorphic to Q/Z and B(R) to Z/2Z, and that there are natural maps from B(Q) to B(Q_p) and to B(R) which embed B(Q) into the direct sum; this natural embedding is not an isomorphism. $\endgroup$
    – Chandan Singh Dalawat
    Commented May 16, 2013 at 4:09
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    $\begingroup$ When I say direct sum, I mean the direct sum of B(R) and B(Q_p) for all primes p. The sum is indexed by the places of Q, not by the natural numbers. $\endgroup$
    – Chandan Singh Dalawat
    Commented May 16, 2013 at 4:13
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    $\begingroup$ Ah, I see. Within this question, I'm just interested in the isomorphism types of Brauer groups, not how they interact with each other (although the latter is definitely the "right" way of looking at things in general); I just want to know what groups are Brauer groups of some field. $\endgroup$
    – Noah Schweber
    Commented May 16, 2013 at 5:55

2 Answers 2

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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.

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The Brauer group of a field $F$ decomposes as the direct sum of its $p$-primary components $\mathrm{Br}(F)_p$, with $p$ prime. A conjecture of Brumer and Rosen (Proc. AMS 19 (1968), 707-711) predicts that for every prime $p$, the component $\mathrm{Br}(F)_p$ is either

  1. trivial,
  2. contains a non-trivial divisible group, or
  3. $p=2$ and $\mathrm{Br}(F)_2$ is an elementary abelian 2-group.

This was proved by Merkurjev (Brauer groups of fields, Comm. Algebra 11 (1983), 2611-2624) when $(F(\mu_p):F)\le2$, as well as by B. Kahn in his Thesis.

Other related papers may be:

T. Wurfel, Ein Frieheitskriterium fur pro-$p$-Gruppen mit Anwendungen auf die Struktur der Brauer-Gruppe, Math. Z. 172 (1980), 81-88

I. Efrat, On fields with finite Brauer groups, Pacific J. Math. (177)(1997), 33-46.

I hope this helps.

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