On the minimality of powers of minimal omega-bounded abelian groups

We describe the structure of totally disconnected minimal omega-bounded abe lian groups by reducing the description to the case of those of them which are subgroups of powers of the p-adic integers Z(p). In this case the descr iption is obtained by means of a functorial correspondence, based on Pontry agin duality, between topological and linearly topologized groups introduce d by Tonolo. As an application we answer the question (posed in Pseudocompa ct and countably compact abelian groups: Cartesian products and minimality, Trans. Amer. Math. Soc. 335 (1993), 775-790) when arbitrary powers of mini mal omega-bounded abelian groups are minimal. We prove that the positive an swer to this question is equivalent to non-existence of measurable cardinal s.