(a) Let $K$ be a number field, and $f$ a monic polynomial whose coefficients are in $\mathcal{O}_{K}$. Let $M$ be a field containing $K$ and $\alpha \in M$. Show that if $f(\alpha)=0$, then $\alpha$ is an algebraic integer.

Hence conclude that if $h \in K[x]$ is monic, with $h^{n} \in \mathcal{O}_{K}[x]$, then $h \in \mathcal{O}_{K}[x]$.

(b) Compute an integral basis for $\mathcal{O}_{\mathbb{Q}(\alpha)}$ when the minimum polynomial of $\alpha$ is $x^{3}-x-4$.