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 affine 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.