Let $\displaystyle \sigma$ be a linear transform on a vector field $\displaystyle V$. Suppose $\displaystyle m(x)$ is the minimal polynomial of $\displaystyle \sigma$, $\displaystyle f(x)$ is a polynomial. Show that if $\displaystyle f(\sigma) $ is invertible, then $\displaystyle m(x),f(x)$ is relatively prime. That is, $\displaystyle (m(x),f(x))=1.$