An odd prime missing the cubic discriminant misses the index

Let $D,E\in\mathbb{Z}$ with $x^3+Dx+E$ irreducible over $\mathbb{Q}$, let $K$ be the resulting cubic field and $\theta\in\mathcal{O}_K$ the root. Let $p$ be an odd prime not dividing the discriminant $\Delta=-4D^3-27E^2$. Then $p$ does not divide the index $[\mathcal{O}_K:\mathbb{Z}[\theta]]$.

This is the index-discriminant relation $\operatorname{disc}(m)=[\mathcal{O}_K:\mathbb{Z}[\theta]]^2\cdot\operatorname{disc}(K)$ specialised to the cubic: since the index squared divides $\Delta$, any prime missing $\Delta$ misses the index.