Finding polynomial orbits.

For a ring $R$, and a polynomial $F\in R[X]$ we consider an $n$-tuple $x_0,x_1,...,x_{n-1}$ of distinct elements of $R$ satisfying $F(x_0)=x_1,F(x_1)=x_2,...,F(x_{n-1})=x_0$, and call it a " cycle " of length $n$. We shall outline a way how to examine which $n$ may appear as the length of a cycle, when $R=Z_K$, i.e. $R$ is the ring of integral elements in a finite extension $K$ of $Q$ (and is the main object of study in algebraic number theory).