I'm working on an exercise from Steven Roman's Advanced Linear Algebra. He asks
Let $\dim(V)<\infty$. If $\tau,\sigma\in\mathscr{L}(V)$, prove that $\sigma\tau=\mathrm{id}$ implies $\tau$ and $\sigma$ are invertible, and that $\sigma=p(\tau)$ for some polynomial $p(x)\in F[x]$, $F$ being the underlying field.
From $\sigma\tau=\mathrm{id}$, $\tau$ must be injective, hence surjective, and thus invertible. Thus $\sigma=\tau^{-1}$, so both $\sigma$ and $\tau$ are invertible. But it is not clear to me how $\sigma$ can be expressed as a polynomial in $\tau$. Can anyone please explain?