{
  "id": "1904.03466",
  "version": "v1",
  "published": "2019-04-06T15:12:16.000Z",
  "updated": "2019-04-06T15:12:16.000Z",
  "title": "A p-adic analogue of Siegel's Theorem on sums of squares",
  "authors": [
    "Sylvy Anscombe",
    "Philip Dittmann",
    "Arno Fehm"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-06T15:12:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "12D15",
      "11S99",
      "11U09"
    ],
    "keywords": [
      "p-adic analogue",
      "siegels theorem",
      "number field",
      "p-adic kochen operator",
      "finite p-pythagoras number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}