What can we say about the minimal polynomial over a field $\mathbb{F}$

Question

Let $\mathbb{F}$ be a field. Suppose $A \in GL(m,\mathbb{F})$, i.e. $A$ is an $m \times m$ invertible matrix with coefficients in $\mathbb{F}$. Now let $A$ have order $n$.

What can we say about the minimal polynomial of $A$?

Answer 1

It is because $A^n=I$, so $A$ is a root of $X^n-1$. The kernel of $F[X]\rightarrow Gl(m,F)$ defined by $P\rightarrow P(A)$ is an ideal which is principal since $F$ is a field, it is generated by the minimal polynomial $P_A$ of $A$ and it contains $X^n-1$, we deduce that $P_A$ divides $X^n-1$.