We study the dynamics of polynomials with coefficients in a non-Archimedean
field $\mathbb{L}$, where $\mathbb{L}$ is the completion of an algebraic
closure of the field of formal Laurent series. We prove that every wandering
Fatou component is contained in the basin of a periodic orbit. We give a
dynamical characterization of polynomials having algebraic Julia sets. More
precisely, we establish that a polynomial with algebraic coefficients (over the
field of formal Laurent series) has algebraic Julia set if and only if every
critical point is non recurrent.