If $\alpha \in \Bbb{C}$ is an algebraic number such that $f(X)\in\Bbb{Z}[X]$ is the minimal polynomial over $\Bbb{Q}$, show that for any prime number $p \in \Bbb{Z}$ we have that $\Bbb{Z}_p[X]/(\bar f(X))$ is isomorphic to $\Bbb{Z}[α]/(p)$, where $(b)$ is the ideal generated by the element $b$ and $\bar f(X)$ is the polynomial in $\Bbb{Z}_p[X]$ that has as coefficients the class of the coefficients of $f(x)$ modulo $p$.
Can anyone give a hint on how to begin?
Show that both rings are isomorphic to $\mathbb Z[X]/(p, f(X))$.