2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/174411Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields.8 pagesAlgebraic GeometryLogicNumber Theory12E30; 14G05; 12F99; 03C60text