Given a monic polynomial $f \in \mathbb{Z}[X]$, I would like to consider the ideal $(f, f')_{\mathbb{Z}[X]} \cap \mathbb{Z}$ in $\mathbb{Z}$. In particular: is it true that this is generated by the discriminant $\Delta(f)$?
I know at least that $\Delta(f)$ is contained in this ideal.