Northcott Property for Bounded-Degree Composita

Definitions and mathematical statement

Fix an algebraic closure $\overline{{\mathbb{Q}}}$ of ${\mathbb{Q}}$. For a positive integer $d$, define \[ {\mathbb{Q}}^{(d)}=\mathop{\mathrm{compositum}}_{\substack{{\mathbb{Q}}\subseteq K\subseteq\overline{{\mathbb{Q}}}\\[K:{\mathbb{Q}}]\leq d}}K, \] the smallest subfield of $\overline{{\mathbb{Q}}}$ containing every such number field $K$. For $\alpha\in\overline{{\mathbb{Q}}}$, let \[ f_\alpha(T)=a\prod_{i=1}^{n}(T-\alpha_i)\in{\mathbb{Z}}[T] \] be its primitive irreducible polynomial, with positive leading coefficient $a$, where $n=[{\mathbb{Q}}(\alpha):{\mathbb{Q}}]$ and the $\alpha_i$ are its complex conjugates. Its absolute multiplicative Weil height is \[ H(\alpha)=\left(a\prod_{i=1}^{n}\max\{1,|\alpha_i|\}\right)^{1/n}. \] A subfield $L\subseteq\overline{{\mathbb{Q}}}$ has Property (N), or the Northcott property, if \[ \forall X\in{\mathbb{R}},\quad X\geq1\ \Longrightarrow\ \#\{\alpha\in L:H(\alpha)<X\}<\infty. \] Question. For a given positive integer $d$, does ${\mathbb{Q}}^{(d)}$ have Property (N)? Equivalently, is \[ \forall X\in{\mathbb{R}},\quad X\geq1\ \Longrightarrow\ \#\{\alpha\in{\mathbb{Q}}^{(d)}:H(\alpha)<X\}<\infty \] true for every positive integer $d$? The source statement does not explicitly specify the range or quantification of $d$; the formulation above uses the natural positive-integer parameter interpretation. The supplied dated literature assessment identifies $d\geq3$ as the open range. This is a supplied status assessment, not an independently verified mathematical or literature conclusion.

Short English statement

For each positive integer $d$, form the field generated by all number fields of degree at most $d$ inside a fixed algebraic closure of ${\mathbb{Q}}$. The question asks whether every fixed height bound contains only finitely many elements of this field. The supplied literature assessment focuses on $d\geq3$; the original statement leaves the range of $d$ implicit.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003242$, source problem code catalogue entry, titled Northcott Property for Bounded-Degree Composita; collection: Oberwolfach Reports Open Problems. No individual proposer is named; the proposed year is recorded as 2016.
• 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 attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source, labeled "Widmer, Northcott property": https://arxiv.org/abs/2310.11337.
• Supplied literature source, labeled "Oberwolfach Reports 2017": https://ems.press/content/serial-article-files/46652.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that Property (N) for ${\mathbb{Q}}^{(d)}$ remains open for $d\geq3$. This is supplied dated curation derived from the OWR record, not independently verified here; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that it was checked against the official Oberwolfach report. That verification claim is supplied metadata, not a verification performed for this draft.