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!