# Square-free discriminants and integral bases

Let $K = \mathbb Q(\alpha)$ is a number field. Suppose $\alpha \in O_K$ and let $f \in \mathbb Z[X]$ be its minimal polynomial. Show that if the discriminant of $f$ is a square-free integer, then $O_K = \mathbb Z[\alpha]$.

I don't really know how to approach this problem. I have observed that:

i) $\mathbb Z[\alpha] \subseteq O_K$ trivially

ii) The problem is equivalent to showing that $\{1,\alpha,\ldots ,\alpha^{n-1} \}$ is an integral basis for $K$

I can't see how the discriminant is at all related, and I suppose this is where my understanding of this topic lacks. Any push in the right direction would be appreciated.

Thanks

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Use the result from your other question math.stackexchange.com/questions/108711 ! –  David Loeffler Feb 13 '12 at 8:41