Indecomposable $p$-local torsion-free groups with quadratic and cubic splitting fields

Indecomposable torsion-free $p$-local Abelian groups of finite rank with quadratic and cubic splitting field $K$ are characterized. As a consequence, for groups with quadratic splitting field $K$ 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.