Closure Conjectures for Infinite Stable Fields

Definitions and mathematical statement

Let $K$ be a field, regarded as a first-order structure in the language intended by the source. No restriction on the characteristic is stated. Its complete theory $T=\operatorname{Th}(K)$ is called stable if no formula $\varphi(x;y)$ has the order property: there are no sequences $(a_i)_{i<\omega}$ and $(b_j)_{j<\omega}$ in a model of $T$ such that $\varphi(a_i;b_j)$ holds exactly when $i<j$.

For context, the standard weight of a complete type $p$ in a stable theory is the supremum of the cardinalities $|I|$ for which there exist a nonforking extension of $p$ over a parameter set $B$, a realization $a$ of that extension, and a family $(b_i)_{i\in I}$ independent over $B$, such that $\operatorname{tp}(a/Bb_i)$ forks over $B$ for every $i\in I$. A type has finite weight when this supremum is finite. This definition alone does not specify what the source means by a field having finite weight: the record does not identify the relevant type or collection of types. The record also does not define strongly stable. These qualifications therefore remain unresolved; they are retained as source predicates in the statements below.

A field $K$ is separably closed if it has no proper finite separable algebraic extension. It is algebraically closed if every nonconstant polynomial in $K[X]$ has a root in $K$.

Conjecture 1, as supplied. For every infinite field $K$, \[ \left(K\text{ is stable and has finite weight in the source's intended sense}\right) \ \Longrightarrow\ K\text{ is separably closed}. \] Conjecture 2, as supplied. For every infinite field $K$, \[ K\text{ is strongly stable in the source's intended sense} \ \Longrightarrow\ K\text{ is algebraically closed}. \] The intended language and the meanings of the two weight-related qualifications require confirmation from the cited sources before these become fully specified first-order model-theoretic conjectures. The displayed implications preserve the supplied targets without resolving those omissions by assumption.

Short English statement

The source proposes two closure conjectures: every infinite stable field of finite weight is separably closed, and every infinite strongly stable field is algebraically closed. The intended language, the field-level finite-weight convention, and the meaning of strongly stable require confirmation. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry, permanent ID $30001403$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied original and clean statements coincide; the registry marks the statement as exact and retained byte-for-byte. No individual proposer is named; the supplied proposed year is 2010.
• Model Theory: Around Valued Fields and Dependent Theories (2010), Oberwolfach Reports, pp. 5--53, DOI 10.4171/owr/2010/01. Supplied source report and literature-source link labelled "original Oberwolfach problem report": https://doi.org/10.4171/owr/2010/01. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Krupiński and Pillay, supplied literature-source attribution: https://doi.org/10.1017/S1474748010000228. No paper title or more specific bibliographic details are supplied.
• Supplied literature assessment, checked on 2026-08-21: the cited paper is reported to prove the weight-one case and to record the finite-weight assertion as Conjecture 0.3; the assessment further reports that later work still treats the general stable-fields assertion as a conjecture. The supplied status is "open". This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.