Open Problems in Mathematical Logic

Problem #13

Stable fields conjecture

Open!!

The stable fields conjecture asserts that every infinite stable field is separably closed.

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Known Partial Results

  • Johnson, Tran, Walsberg and Ye showed that large stable fields are separably closed [JTW23].

Notes

Macintyre proved that every infinite ω\omega-stable field is algebraically closed [Mac71]. Cherlin and Shelah extended the result to infinite superstable fields [CS80]. Wood proved that all separably closed fields are stable [Woo79].

Reference for the problem statement

[HHJ19]Yatir Halevi, Assaf Hasson, and Franziska Jahnke, A conjectural classification of strongly dependent fields, Bulletin of Symbolic Logic, 2019 [link] [doi]

Additional References

[Mac71]Macintyre, On ω-categorical theories of fields, Fundamenta Mathematicae, 1971

[Woo79]Wood, Notes on the stability of separably closed fields, Journal of Symbolic Logic, 1979

[CS80]Cherlin and Shelah, Superstable fields and groups, Annals of Mathematical Logic, 1980

[JTW23]Johnson, Tran, Walsberg, and Ye, The Étale-open topology and the stable fields conjecture, Journal of the European mathematical society, 2023

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