Let $P\in\mathbb{F}_p[T]$ (not supposed irreducible). All roots $\xi$ of $P$ have a certain order $k$ such that $\xi^k=1$.
Question: is it possible to know the order of the roots of the given polynomial $P$, or at least a upper bound of the order?
It is clear that if $k$ is an order of (a root of) $P$ then by the description of factors of cycltomic polynomials $\Phi_k$ then
$$ \text{ord}_k(p)\leq \deg(P) $$
where $\text{ord}_k(p)$ is the order of $p$ in $(\mathbb{Z}/k\mathbb{Z})^*$ and that basically $$ \log_p(k)\leq \text{ord}_k(p) $$ because for $a$ to be the order of $p$ we must have $p^a\geq k$. So we have $$ \log_p(k)\leq\deg(P) $$ and hence $$ k\leq p^{\deg(P)} $$
This is a very big upper bound and for algorithms I would like to have a smaller bound or better a list of possible orders.
Thanks for your help!