Let $f(X)$ be a polynomial with integer coefficients such that $f : \mathbb{Z} \longrightarrow \mathbb{Z}$ is onto. Show that $f(X)=\pm X+c$ for some integer $c$.
I was given this question in a lecture on irreducible polynomials. I think a solution using Ostrowski's criterion was presented, but I couldn't quite follow every step. Could someone help? Thanks
Hint: For $x \gg 1$, we have that $f(x) \approx x^{\deg f}$. For $f: \mathbb{Z} \to \mathbb{Z}$, this can't possibly be onto.