Time-domain description of behaviors over finite fields

We consider autonomous behaviors over a finite field with characteristic values that do not necessarily belong to the field. The time domain description of the behavior is given in a suitable field extension of the base field. The problem that we consider is how to derive a description completely within the base field. For the case of behaviors over the reals there is a common splitting field for all irreducible polynomials, the complex field. Complex trajectories induce real trajectories by restricting coefficients of complex conjugate exponentials to be complex conjugate as well. For the case of finite fields the situation is more complicated as there does not exist a single finite field extension in which all polynomials over the base field split. In this paper we describe a systematic procedure to obtain explicit expressions for all trajectories in the behavior whose components take values in the base field.