Ramification indices and residue class degrees of $\mathfrak{O}_K$ where $K=Q[\alpha]$, $f(\alpha)=0$, and $f(x)=x^3-x-1$.

Question

I already know that $\alpha^3-\alpha-1=0$ implies that $\{1,\alpha, \alpha^2 \}$ creates an integral basis for $\mathbb{A} \cap \mathbb{Q}[\alpha]$. I'd like to try to use Dedekind's theorem in some fashion to find the ramification indices and residue class degrees of the prime ideals of $\mathfrak{O}_K$ which lie above 2, 5, and 23.

I'm not sure if it would be an entirely separate problem for each of the primes, but any guidance would be much appreciated.