  We study the dynamics of polynomials with coefficients in a non-Archimedean
field $K,$ where $K$ is a field containing a dense subset of algebraic elements
over a discrete valued field $k.$ We prove that every wandering Fatou component
is contained in the basin of a periodic orbit. We obtain a complete description
of the new Julia set points that appear when passing from $K$ to the Berkovich
line over $K$. We give a dynamical characterization of polynomials having
algebraic Julia sets. More precisely, we establish that a polynomial with
algebraic coefficients has algebraic Julia set if every critical element is
nonrecurrent.
