Minimal polynomial question

Question

If I have an $n \times n$ matrix, all elements of which are defined over $\mathbb{Z}n$, and we have a min polynomial for this matrix, $p$. The coefficients of this polynomial are as follows: $a_0$, $a_1$,..$a_L$. $L$ is degree of the minimal polynomial. Can we safely assume all $a_0$, $a_1$,..$a_L$ <= m?

Answer 1

The minimal polynomial of a matrix $M$ over a field $\mathbb{F}$ is the monic polynomial $p(x)$, with coefficients in $\mathbb{F}$, of least degree satisfying $p(M)=0$.

So if your matrix has elements in $\mathbb{Z}_p$, the minimal polynomial's coefficients are also elements of the same finite field. And of course you can represent every equivalence class $[a_i]=\{a_i + kp\,\vert\, k\in\mathbb{Z}\}$ using an integer between $0$ and $p-1$ (or $1$ and $p$, if you prefer.)