On the Baer–Kaplansky theorem for torsion free groups with quadratic splitting fields

The connection between the structure of abelian group and the structure of endomorphism ring is a classic question in abelian group theory. In particular, Baer and Kaplansky proved that this connection is very strong for torsion groups: two abelian torsion groups are isomorphic if and only if their endomorphism rings are isomorphic. In more general cases for torsion-free and mixed abelian groups the Baer–Kaplansky theorem is not true.
This paper deals with the class of $p$-local torsion-free abelian groups of finite rank. Let $K$ be a field such that $\mathbb{Q}\subset K\subset\widehat{\mathbb{Q}}_p$ and let $R=K\cap\widehat{\mathbb{Z}}_p,$ where $\widehat{\mathbb{Z}}_p$ is the ring of $p$-adic integers, $\widehat{\mathbb{Q}}_p$ is the field of $p$-adic numbers, $\mathbb{Q}$ is the field of rational numbers. We say that $K$ is a splitting field ($R$ is a splitting ring) for a $p$-local torsion-free reduced group $A$ or that a group $A$ is $K$-decomposable group if $A\otimes_{\mathbb{Z}_p}R$ is the direct sum of a divisible $R$-module and a free $R$-module. Torsion-free $p$-local abelian groups of finite rank with quadratic splitting field $K$ are characterized. As an application it is proved that $K$-decomposable $p$-local torsion free abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.