2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69660We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.9 pages, uses a4paper, calrsfsNumber TheoryLogic03B25, 11U05text