Could someone please explain to me why
$ \mathbb{F}_p [X] / \langle\bar{f_\alpha} (X)\rangle \,\ \cong \,\ \mathbb{Z}[X] / \langle p, f_\alpha (X) \rangle \,\ \cong \,\ \mathbb{Z} [\alpha] / \langle p\rangle $?
Where $p$ is prime in $\mathbb{Z}$, $ \bar{f_\alpha} $ is the polynomial obtained by taking the coefficients of $ f_\alpha $ modulo p (and $ f_\alpha $ is the minimal polynomial of $\alpha$ in $ \mathbb{Z}[X] $) I know that $ \mathbb{F_p} \cong \mathbb{Z}/p\mathbb{Z} $.
Thanks.