Compact groups with a dense free abelian subgroup

The compact groups having a dense infinite cyclic subgroup (known as monothetic compact groups) have been studied by many authors for their relevance and nice applications. In this paper we describe in full details the compact groups K with a dense free abelian subgroup F and we describe the minimum rank r_t(K) of such a subgroup F of K. Surprisingly, it is either finite or coincides with the density character d(K) of K.