  Building 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 $\xi$ 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.
