On determining the ring of integers of a cubic number field generated by a root of $x^3-x+1$

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.

Solution 1:

If $A$ is a $\mathbb Z$-subalgebra of $\mathcal O_K$ which spans $K$ as a vector space, then the index $[\mathcal O_K:A]$ is finite, and the discriminant of $A$ is equal to $[\mathcal O_K:A]^2$ times the discriminant of $\mathcal O_K$ (which is to say, the discriminant of the number field $K$).

So a sensible thing to do, when confronted with a problem like yours, is to begin by computing the discriminant of $A$, since this will give information about the possible values of $[\mathcal O_K:A]$.

When $A = \mathbb Z[\alpha]$, the discriminant of $A$ is just the discriminant of the minimal polynomial of $\alpha$, and so is particularly straightforward to compute.