Brauer groups are not characterized by Ulm invariants

Two important invariants of a fieldF are its Brauer groupB(F) and its character groupX(F). IfF is countable, these are countable abelian torsion groups, and so are determined by their Ulm invariants. We show here that Ulm’s invariants do not determine Brauer groups or character groups of uncountable fields. An essential tool, which is entirely group theoretic in nature, is a fact about ultraproducts of torsion groups. Supported in part by NSF Grant No. DMS-8500883. Supported in part by NSA Grant No. MDA904-85-H-0014. Supported in part by NSF Grant No. DMS-8500929.