2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64500Building on work of Davenport and Schmidt, we mainly prove two results. The first one is a version of Gel'fond's transcendence criterion which provides a sufficient condition for a complex or $p$-adic number $ξ$ to be algebraic in terms of the existence of polynomials of bounded degree taking small values at $ξ$ together with most of their derivatives. The second one, which follows from this criterion by an argument of duality, is a result of simultaneous approximation by conjugate algebraic integers for a fixed number $ξ$ that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.The section 4 of this new version has been rewritten to simplify the proof of the main result. Other results in Sections 9 and 10 have been improved. To appear in Compositio MathNumber Theory11J13text