  We use the theory of resultants of polynomials to study the stability of an
arbitrary polynomial over a finite field, that is, the property of having all
its iterates irreducible. This result partially generalises the quadratic
polynomial case described by R. Jones and N. Boston. Moreover, for
characteristic three, we show that certain polynomials of degree three are not
stable. We also use the Weil bound for multiplicative character sums to
estimate the number of stable arbitrary polynomials over finite fields of odd
characteristic.
