## Abstract

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 of 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 of characteristic 0 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.

Original language | English |
---|---|

Article number | 108408 |

Journal | Advances in Mathematics |

Volume | 404 |

DOIs | |

State | Published - 6 Aug 2022 |

### Bibliographical note

Funding Information:The first author was supported by the Kreitman foundation fellowship and the Fields Institute for Research in Mathematical Sciences . The third author was partially supported by Israel Science Foundation grant number 290/19 .

Publisher Copyright:

© 2022 Elsevier Inc.

## Keywords

- ACVF
- dp-Minimal
- Interpretable field
- p-Adically closed
- RCVF
- Valued field

## ASJC Scopus subject areas

- Mathematics (all)