Diophantine approximation by conjugate algebraic integers

Building on the 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 xi to be algebraic in terms of the existence of polynomials of bounded degree taking small values at xi 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 xi that is either transcendental or algebraic of sufficiently large degree. We also present several constructions showing that these results are essentially optimal.