2021/09/01 by Yatir Halevi, Halevi, Yatir, Assaf Hasson +3
Mathematics · #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2109.00569
openalex publication_date 2021/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 ℚp is definably isomorphic to a finite extension of ℚp, 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.