The decomposability problem for torsion-free abelian groups is analytic complete

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is ${\mathrm{\Sigma }}_3^0$-complete. However, when we consider groups of infinite rank, it becomes ${\mathrm{\Sigma }}_1^1$-complete, so it cannot be characterized by a first-order formula in the language of arithmetic.