Stable polynomials over finite fields

We use the theory of resultants to study the stability, that is, the property of having all iterates irreducible, of an arbitrary polynomial $f$ over a finite field $\mathbb{F}_q$. This result partially generalizes the quadratic polynomial case described by R. Jones and N. Boston. Moreover, for $p=3$, 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 polynomials over a finite field of odd characteristic.