Finding the cycle set of specific polynomial I have a school assignment where the problem lies in finding the cycle set of two different polynomials.

I tried to look up different tools on how to solve these problems, but I have a hard time finding it.

If anyone were so kind to help me, solve but also, understand the way to figuring things like this out.

The problem states:

Determine the cycle set for these polynomials:

$x^4 + x^2 + 1$ over $F_2$.

$x^3 + x + 1$ over $F_3$.

Thanks!
This is quite combinatorial in nature and the suggestion in the comment is exactly how one would do small examples. General results are much more complicated for arbitrary polynomials, but simple for primitive polynomials.

If the generating polynomial $f$ of degree $d$ over $F_q$ is primitive (i.e., $f$ is irreducible, and has a root in the splitting field of order $q^d-1,$) then the cycle length is maximal, and equal to $q^d-1$ for any nonzero loading, and is otherwise a single cycle at zero.

If the generating polynomial $f$ is not irreducible but is a product of irreducibles, the cycle structure can get quite messy.

Most generally, form the companion matrix $C_f$ of the polynomial $f$ of degree $d,$ which is a $d\times d$ polynomial over $F_q.$ Then the length of any cycle that can be generated by the given degree $d$ polynomial divides the order of $C_f$ in the general linear group $GL(F_q,d)$ of $d\times d$ invertible matrices where the group operation is matrix multiplication. This means that the length of any cycle will be a divisor of 
$$
q^{k(k-1)/2}(q-1)(q^2-1)\cdots (q^d-1)
$$
and this is all we can say in general.