2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/168783Let R be a recursive subring of a number field. We show that recursively enumerable sets are diophantine for the polynomial ring R[Z].Previous version had a mistake in Proposition 18. This problem is avoided by working only with number fields instead of finitely generated fields of characteristic zeroNumber TheoryLogic11D99 (Primary) 03D25, 12L12, 11R09, 12E10 (Secondary)text