Northcott Property for Totally Real Integer Rings

Definitions and mathematical statement

The supplied question is whether the ring $\mathcal O_{\mathbb Q_{\mathrm{tr}}^{(d)}}$ has Property $(N)$. The record identifies $\mathbb Q_{\mathrm{tr}}^{(d)}$ as the maximal totally real subfield of $\mathbb Q^{(d)}$, but does not define $\mathbb Q^{(d)}$, specify the range or quantification of $d$, or define Property $(N)$. Consequently, the following explicit formulation is conditional on conventions that require source verification.

Fix an algebraic closure $\overline{\mathbb Q}\subset\mathbb C$. An algebraic number is totally real if every one of its conjugates over $\mathbb Q$ lies in $\mathbb R$. Write $\mathbb Q_{\mathrm{tr}}$ for the field of all totally real algebraic numbers. Under the conventional interpretation that, for a fixed positive integer $d$, $\mathbb Q^{(d)}$ is the compositum in $\overline{\mathbb Q}$ of all number fields of degree at most $d$, set \[ K_d=\mathbb Q^{(d)}\cap\mathbb Q_{\mathrm{tr}},\qquad \mathcal O_{K_d}=\{\alpha\in K_d:\alpha\text{ satisfies a monic polynomial in }\mathbb Z[x]\}. \] Thus $K_d$ is the maximal totally real subfield described in the record.

For an algebraic number $\alpha$ with primitive irreducible polynomial $a\prod_{i=1}^{n}(x-\alpha_i)\in\mathbb Z[x]$, where $a>0$, define its absolute multiplicative Weil height by \[ H(\alpha)=\left(a\prod_{i=1}^{n}\max\{1,|\alpha_i|\}\right)^{1/n}. \] If Property $(N)$ means the Northcott property for this height, the question becomes: does \[ \forall B\in\mathbb R_{>0},\qquad \#\{\alpha\in\mathcal O_{K_d}:H(\alpha)\le B\}<\infty \] hold for the intended values of $d$? No bound on the degrees of the elements $\alpha$ is imposed in this formulation.

A different possible interpretation uses the house $\max_{1\le i\le n}|\alpha_i|$. Under that interpretation the question instead asks whether \[ \forall B\in\mathbb R_{>0},\qquad \#\left\{\alpha\in\mathcal O_{K_d}:\max_{1\le i\le n}|\alpha_i|\le B\right\}<\infty. \] The supplied record does not specify which boundedness condition defines Property $(N)$; these formulations are not identified with one another here. The intended definition and quantification of $d$ remain unresolved.

Short English statement

The source asks whether the algebraic integers in the maximal totally real subfield of $\mathbb Q^{(d)}$ have Property $(N)$. A precise interpretation requires checking the definition of $\mathbb Q^{(d)}$, the intended values of $d$, and whether Property $(N)$ concerns bounded Weil height or bounded house. The supplied open status is dated curation, not an independently verified conclusion.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30003244$, title "Northcott Property for Totally Real Integer Rings," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2016 and names no individual proposer.
• Supplied report citation: Definability and Decidability Problems in Number Theory (2017), Oberwolfach Reports, pp. 2793--2866, DOI 10.4171/owr/2016/49. https://doi.org/10.4171/owr/2016/49. The background describes these reports as documenting mathematical workshops and their open-problem discussions; no precise location within the report is supplied for this question.
• Supplied literature source, labelled "Oberwolfach Reports 2017": https://ems.press/content/serial-article-files/46652.
• Supplied literature source, labelled "Daans et al., Universal quadratic forms": https://doi.org/10.1112/jlms.70022. No further bibliographic details or supporting passage are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the question open and states that the totally real integer ring is still treated as an open Northcott-property question. This is supplied dated curation derived from the OWR record, not independently verified literature assessment; the record itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a prior manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves its target. These are supplied registry assertions, not mathematical verification of this draft or resolution of its missing definitions.