Index of a number field (awaiting review)

If $K$ is a number field with ring of integers $\mathcal{O}_K$, then for all $\alpha\in\mathcal{O}_K$ such that $K=\Q(\alpha)$, the index of $\alpha$, $i(\alpha)$ is the index of the order $\Z[\alpha]$.

The index of the number field is the greatest common divisor of all $i(\alpha)$ with $\alpha$ as above.