What does it means to say that a complex number $\alpha$ is algebraic over $\mathbb{Q}$ ? Define the minimal polynomial of $\alpha$.

Suppose that $\alpha$ satisfies a nonconstant polynomial $f \in \mathbb{Z}[X]$ which is irreducible over $\mathbb{Z}$. Show that there is an isomorphism $\mathbb{Z}[X] /(f) \cong \mathbb{Z}[\alpha]$.

[You may assume standard results about unique factorisation, including Gauss's lemma.]