If $p$ misses the index $[\mathcal{O}_K:\mathbb{Z}[\theta]]$ it misses the exponent of $\theta$

Let $K$ be a number field and $\theta\in\mathcal{O}_K$. If a prime $p$ does not divide the index of $\mathbb{Z}[\theta]$ as an additive subgroup of $\mathcal{O}_K$, then $p$ does not divide RingOfIntegers.exponent $\theta$ — the smallest positive integer $d$ with $d\,\mathcal{O}_K\subseteq\mathbb{Z}[\theta]$.

This is the bridge that converts an index hypothesis into the exact hypothesis Mathlib's Kummer-Dedekind bijection requires. It follows from the fact that the exponent divides the index, which is Lagrange applied to the conductor.