  We consider in this paper the rigorous justification of the
Zakharov-Kuznetsov equation from the Euler-Poisson system for uniformly
magnetized plasmas. We first provide a proof of the local well-posedness of the
Cauchy problem for the aforementioned system in dimensions two and three. Then
we prove that the long-wave small-amplitude limit is described by the
Zakharov-Kuznetsov equation. This is done first in the case of cold plasma; we
then show how to extend this result in presence of the isothermal pressure term
with uniform estimates when this latter goes to zero.
