Integer Predicates in Models of Totally Real Algebraic Integers

Definitions and mathematical statement

An algebraic integer is an algebraic number satisfying a monic polynomial in ${\mathbb{Z}}[x]$. It is totally real if all its conjugates over ${\mathbb{Q}}$ are real. Let $A$ be the ring of totally real algebraic integers intended by the source, containing the distinguished integer subring ${\mathbb{Z}}$. The supplied record does not determine whether $A$ is the full ring of all totally real algebraic integers or a particular subring.

Interpret "${\mathbb{Z}}$ named" using the language of rings expanded by a unary predicate $P$. Write $(A,{\mathbb{Z}})$ for the structure in which $P$ denotes ${\mathbb{Z}}$, and let \[ (R,S')\models\operatorname{Th}(A,{\mathbb{Z}}), \] where $\operatorname{Th}(A,{\mathbb{Z}})$ is the complete first-order theory of this expanded structure and $S'\subseteq R$ interprets $P$.

For clarity, an ultrapower presentation would consist of a nonempty index set $I$, an ultrafilter $\mathcal U$ on $I$, and a specified ring isomorphism \[ \iota:A^I/\mathcal U\longrightarrow R. \] The predicate induced by the distinguished integers would then be \[ S=\iota\!\left(\left\{[a_i]_{\mathcal U}:\{i\in I:a_i\in{\mathbb{Z}}\}\in\mathcal U\right\}\right). \] This describes how $S$ is defined if the source's "associated ultrapower model" is an ultrapower of $(A,{\mathbb{Z}})$ identified with the ring $R$.

Source question, subject to the unresolved association. For $(R,S')$ as above and its source-intended associated ultrapower expansion $(R,S)$, must \[ S'\subseteq S, \qquad\text{equivalently}\qquad \forall r\in R\;(r\in S'\Longrightarrow r\in S)? \] The supplied record does not specify the construction of the associated ultrapower, the identification of its underlying ring with $R$, or any additional compatibility required between $S'$ and $S$. Consequently, it does not determine a fully quantified conjecture over ultrapower presentations. In particular, the description above does not assert that every model of $\operatorname{Th}(A,{\mathbb{Z}})$ is itself an ultrapower or that the proposed inclusion holds for every possible presentation. These qualifications require clarification from the original mathematical context.

Short English statement

The question asks whether an integer predicate $S'$ in a model of the theory of $(A,{\mathbb{Z}})$ must be contained in the integer predicate $S$ supplied by an associated ultrapower on the same underlying ring. The meaning and hypotheses of this association are missing, so the intended quantifiers remain unresolved. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003257$, title "Integer Predicates in Models of Totally Real Algebraic Integers," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2016$ and supplies no named proposer.
• Supplied report citation: Definability and Decidability Problems in Number Theory (2017), Oberwolfach Reports, pp. 2793--2866, DOI 10.4171/owr/2016/49. Supplied source link: https://doi.org/10.4171/owr/2016/49.
• 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 conjecture $S'\subseteq S$ unresolved and reports that the source explicitly recorded that no further implication was known. This is supplied dated curation, not an independently verified literature assessment or independently checked quotation. The record itself requests independent verification of time-sensitive status and cited claims.