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Preprint Number 2042

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2042. Yatir Halevi and Assaf Hasson and Ya'acov Peterzil
Interpretable Fields in Various Valued Fields

Submission date: 1 September 2021


Let K=(K,v, ...) be a dp-minimal expansion of a non-trivially valued field of characteristic 0 and F an infinite field interpretable in K.
Assume that K is one of the following: (i) V-minimal, (ii) power bounded T-convex, or (iii) P-minimal (assuming additionally in (iii) generic differentiability of definable functions). Then F is definably isomorphic to a finite extension K or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in ℚ_p is definably isomorphic to a finite extension of ℚ_p, answering a question of Pillay's.
Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if K is an infinite dp-minimal pure field then every field definable in K is definably isomorphic to a finite extension of K.
The proof avoids elimination of imaginaries in K replacing it with a reduction of the problem to certain distinguished quotients of K.

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Full text arXiv 2109.00569: pdf, ps.

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