Stable Fields Conjecture

The stable fields conjecture predicts that an infinite field whose complete first-order theory in the language of rings is stable must be separably closed. Since separably closed fields are known to be stable, the conjecture would characterize exactly which infinite fields have stable theories. It grew out of Macintyre's 1971 theorem that every infinite $\omega$-stable field is algebraically closed [Macintyre1971Fields], and the general question dates from about that time rather than from a single proposal.

Cherlin and Shelah extended Macintyre's theorem to the superstable case: infinite superstable fields are algebraically closed [CherlinShelah1980]. The conjecture is also known for stable fields of finite weight in selected ranks, and a substantial recent advance of Johnson, Tran, Walsberg and Ye establishes it for large stable fields via the étale-open topology [JohnsonTranWalsbergYe2022]. Related work explores the parallel landscape of NIP fields [DittmannEtAl2023NIP].

For arbitrary stable fields no classification is available: a proof would have to handle stable fields carrying no largeness, rank, or weight hypotheses, where the existing algebraic and topological techniques do not yet reach. The conjecture remains open.