Northcott Property for Rings of Integers in Bounded-Degree Composita

Definitions and mathematical statement

Fix an algebraic closure $\overline{\mathbb Q}\subset\mathbb C$. For an integer $d>2$, define \[ \mathbb Q^{(d)}=\mathop{\mathrm{compositum}}_{\substack{\mathbb Q\subseteq K\subseteq\overline{\mathbb Q}\\ [K:\mathbb Q]\le d}}K, \] the smallest subfield of $\overline{\mathbb Q}$ containing every such number field $K$. Define its ring of integers by \[ \mathcal O_{\mathbb Q^{(d)}}=\{\alpha\in\mathbb Q^{(d)}:\alpha\text{ satisfies a monic polynomial in }\mathbb Z[x]\}. \] For $\alpha\in\overline{\mathbb Q}$, write its primitive irreducible polynomial over $\mathbb Z$, with positive leading coefficient, as \[ f_\alpha(x)=a\prod_{j=1}^{n}(x-\alpha_j),\qquad a>0. \] Its absolute logarithmic Weil height is \[ h(\alpha)=\frac{1}{n}\left(\log a+\sum_{j=1}^{n}\log\max\{1,|\alpha_j|\}\right). \] A subset $S\subseteq\overline{\mathbb Q}$ has Property $(N)$, also called the Northcott property, if for every real $B\ge0$, the set $\{\alpha\in S:h(\alpha)\le B\}$ is finite. Using the multiplicative height $H(\alpha)=\exp(h(\alpha))$ gives the same property.

Question. Is it true that for every integer $d>2$ and every real $B\ge0$, \[ \#\{\alpha\in\mathcal O_{\mathbb Q^{(d)}}:h(\alpha)\le B\}<\infty? \] Here the integer domain for $d$ is the conventional interpretation of the supplied degree cutoff; the record writes only $d>2$.

Unresolved source qualification. The displayed question follows the supplied original and cleaned statements. The supplied literature assessment instead uses the notation $\mathcal O_{\mathbb Q_{\mathrm{tr}}^{(d)}}$, without defining $\mathbb Q_{\mathrm{tr}}^{(d)}$ or identifying it with $\mathbb Q^{(d)}$. Consequently, the intended relationship between that assessment and the displayed question remains unresolved. No totally real restriction has been added to the displayed formulation.

Short English statement

For each integer $d>2$, form the compositum of all number fields of degree at most $d$. The question asks whether only finitely many algebraic integers in this field have height below any prescribed bound. The supplied literature assessment uses a different, undefined field notation, so its applicability to this formulation requires review.

Sources

• Local UnsolvedMath problem registry, supplied complete record with permanent ID $30003118$, source problem code catalogue entry, titled Northcott Property for Rings of Integers in Bounded-Degree Composita; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied report citation: Diophantische Approximationen (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/21, pp. 1099--1187. Supplied source link: https://doi.org/10.4171/owr/2016/21. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation's 2017 label and its 2016 DOI component are retained as supplied.
• Supplied literature source labeled "OWR report": https://ems.press/content/serial-article-files/46625?nt=1.
• Supplied literature source labeled "Bombieri--Zannier": https://arxiv.org/abs/1205.2491. No further bibliographic details or specific supporting result are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports that the question for $\mathcal O_{\mathbb Q_{\mathrm{tr}}^{(d)}}$ is explicitly open in the record and that no general affirmative result was located. This is supplied dated literature triage, not independently verified here; its time-sensitive status, cited claims, and applicability to $\mathbb Q^{(d)}$ require independent verification.
• The registry also supplies the statement label corrected_verified, explained as preserving mathematical content while repairing notation, typography, or omitted definitions for self-contained publication. This is supplied metadata, not independent mathematical verification of this draft or its sources.