{
  "id": "2109.00569",
  "version": "v1",
  "published": "2021-09-01T18:41:52.000Z",
  "updated": "2021-09-01T18:41:52.000Z",
  "title": "Interpretable Fields in Various Valued Fields",
  "authors": [
    "Yatir Halevi",
    "Assaf Hasson",
    "Ya'acov Peterzil"
  ],
  "categories": [
    "math.LO",
    "math.AC"
  ],
  "abstract": "Let $\\mathcal{K}=(K,v,\\ldots)$ be a dp-minimal expansion of a non-trivially valued field of characteristic $0$ and $\\mathcal{F}$ an infinite field interpretable in $\\mathcal{K}$. Assume that $\\mathcal{K}$ 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 $\\mathcal{F}$ is definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its residue field. In particular, every infinite field interpretable in $\\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\\mathbb{Q}_p$, answering a question of Pillay's. Using Johnson's work on dp-minimal fields and the machinery developed here, we conclude that if $\\mathcal{K}$ is an infinite dp-minimal pure field then every field definable in $\\mathcal{K}$ is definably isomorphic to a finite extension of $K$. The proof avoids elimination of imaginaries in $\\mathcal{K}$ replacing it with a reduction of the problem to certain distinguished quotients of $K$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-01T18:41:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "valued field",
      "interpretable fields",
      "finite extension",
      "definably isomorphic",
      "infinite field interpretable"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}