Consider a set $M_{m\times m}(F)$ of matrices of order $m\times n$ over the finite field $\mathbb{F_p}$. The set $M_{m\times m}(\mathbb{F_p})$ forms a $Ring$ under the binary operations (addition, multiplication) $\bigoplus_p$ and $\bigotimes_p$ respectively. Also the set $M_{m\times m}(F)$ forms a vector space over the finite field $\mathbb{F_p}$.
Suppose, $X$ is a square matrix belongs to the $M_{m\times m}(F)$. Find out the minimal polynomial of $X$ (whose coefficients are from the finite field $\mathbb{F_p}$. Determine the smallest degree of the characteristic polynomial such that the the minimal polynomial of the matrix $X$ would divide the characteristic polynomial of X.