From: victor@idaccr.org (Victor S. Miller)
Subject: Re: Brauer group of a field
Date: 20 Feb 2001 20:14:54 -0500
Newsgroups: sci.math.research
Summary: Which abelian groups can be Brauer groups of some field?

>>>>> "Alan" == Alan Gross <alan_gross@my-deja.com> writes:

Alan> Is it true that for every finite Abelian group G it is possible
Alan> to find a field K such that the Brauer group of K isomorphic to
Alan> G ?

Alan> Thanks, Alan

I got the following information from Al Hales:

Not all finite groups can be Brauer Groups.  For instance the cyclic group of order three
does not occur (Merkurjev).  But any divisible torsion groups can
occur (also Merkurjev).  The Ulm length of Br(K) can be as large as
Omega^3 (Fein-Schacher), but it is also not known if it can be
arbitrarily large.  Also (Fein-Schacher) any torsion abelian group can
occur as Br(L/K), the subgroup of Br(K) split by L, for some K<L.

Hales paper shows, for instance, that any direct sum of cyclic
3-groups occurs as a direct summand of some Br(K).  This is
interesting in view of Merkurjev's first result above.  His paper also
shows that Brauer groups need not be simply presented, and indeed need
not be determined by their Ulm invariants.  Some of this is joint work
with Hales Fein and Schacher.

I hope I've given you good pointers.  You might also start with two
good surveys of this in AMS Contemporary Mathematics volume 13, a
conference in honor of Nathan Jacobson.  One article is by Amitsur and
the other by Schacher.

References:

90j:12010 Hales, Alfred W. Abelian groups as Brauer and character
   groups. Abelian group theory (Oberwolfach, 1985), 487--507,
   Gordon and Breach, New York, 1987.

88d:12009 Fein, B.; Hales, A.; Schacher, M. Brauer groups are
   not characterized by Ulm invariants. Israel J. Math. 55 (1986),
   no. 3,
   363--379.

85f:12006 Merkurjev, A. S. Brauer groups of fields. Comm.
   Algebra 11 (1983), no. 22, 2611--2624.
-- 
Victor S. Miller     | " ... Meanwhile, those of us who can compute can hardly
victor@idaccr.org    | be expected to keep writing papers saying 'I can do the
CCR, Princeton, NJ   | following useless calculation in 2 seconds', and indeed
    08540 USA        | what editor would publish them?"  -- Oliver Atkin