Suppose that $\alpha$ is a zero of $x^{3}-x+3$ and that $K=\mathbb{Q}(\alpha)$. Show that $[K: \mathbb{Q}]=3$. Show that $O_{K}$, the ring of integers in $K$, is $O_{K}=\mathbb{Z}[\alpha]$.

[You may quote any general theorem that you wish, provided that you state it clearly. Note that the discriminant of $x^{3}+p x+q$ is $-4 p^{3}-27 q^{2}$.]