I have the following question:
Let $\alpha$ be a root of the polynomial $f(x) = x^3-x+1$, and let $K = \mathbb{Q}(\alpha)$. Show that $\mathcal{O}_{K} = \mathbb{Z}[\alpha]$.
As I understand it, I need to show that $\{1, \alpha, \alpha^{2}\}$ form a $\mathbb{Z}$-basis for $\mathcal{O}_{K}$, but it is not clear what a good method for that is.