Let $A\in M_n(\mathbb C)$ be a square matrix of order $n$. Suppose that the characteristic polynomial of $A$ equals its minimum polynomial. It is well known that every matrix that commutes with $A$ is a polynomial in $A$.
Suppose that $A$ has integral elements (that is $a_{ij}\in\mathbb Z$) and $B$ is a matrix also with integral elements that commutes with $A$. So, $B$ is of the form
$B=c_0+c_1A+\cdots +c_kA^k.$
Can we conclude that $c_i\in\mathbb Z$ for $i=1,\ldots ,k$?