Let $K$ be a number field and let $\mathcal{O}_K$ be its ring of integers. For $a \in \mathcal{O}_K$ not contained in any proper subfield of $K$, the ring $\mathbb{Z}[a]$ is contained in $\mathcal{O}_K$ and has the same rank as $\mathcal{O}_K$ as a $\mathbb{Z}$-module. Define the minimal index of $K$ by

$$\iota(K) = \min_{\substack{a \in \mathcal{O}_K \\ a \text{ not in any proper subfield of }K}} \text{disc}(\mathbb{Z}[a])/\text{disc}(\mathcal{O}_K).$$

Note that the minimum exists because it is taken over a set of positive integers.

For fields of small degree over $\mathbb{Q}$ (say $[K:\mathbb{Q}] \leq 5$), are there any expectations for the sum

$$\displaystyle \sum_{K : \text{disc}(K) \leq X} \iota(K)?$$

That is, are there conjectures (or better yet, results) on the size of this sum?

  • The usual index of an element will be $\sqrt{\text{disc}(\mathbb{Z}[a])/\text{disc}(\mathcal{O}_K)}$. @alpoge By Minkowski arguments, this is bounded above by $|\text{disc}(\mathcal{O}_K)|^{(d-2)/2}$, with some improvements when d=3 or 5, see "Algebraic Integers with Small Discriminant" by Thunder and Wolfskill. – Zack Wolske Dec 6 at 21:28

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