On the torsion rank of divisible multiplicative groups of fields

In his characterization of the divisible abelian groups which are isomorphic to the unit group of a field, Greg Oman asked the following question: if G is a divisible abelian group that can be realized as the multiplicative group of a field, must the torsion rank of G be either 0 or infinite? The purpose of this paper is to provide a positive answer to this question.