2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172737We develop a theory of arithmetic Newton polygons of higher order, that provides the factorization of a separable polynomial over a $p$-adic field, together with relevant arithmetic information about the fields generated by the irreducible factors. This carries out a program suggested by Ø. Ore. As an application, we obtain fast algorithms to compute discriminants, prime ideal decomposition and integral bases of number fields.In this version we correct some minor mistakesNumber Theory11S15; 11R04; 11R29text