{
  "id": "2411.07880",
  "version": "v1",
  "published": "2024-11-12T15:34:54.000Z",
  "updated": "2024-11-12T15:34:54.000Z",
  "title": "On Classifying Extensions of $p$-adic Fields",
  "authors": [
    "Shreya Dhar",
    "River Newman",
    "Grayson Plumpton",
    "Chenglu Wang"
  ],
  "comment": "20 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime and let $\\mathbb{Q}_p$ be the field of $p$-adic numbers. It is known that the finite extensions of $\\mathbb{Q}_p$ of a given degree are finite up to isomorphism. Given a cubic field extension $L$ of $\\mathbb{Q}_p$ generated by the root of an irreducible polynomial $h$, we present a practical (closed-form) method to determine the isomorphism class in which $L$ lives, based on the coefficients of $h$. We discuss the subtleties of the wildly ramified case, when the degree of the extension coincides with $p$, the characteristic of the residue field. We also present a method for tamely ramified extensions of arbitrary prime degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-12T15:34:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11S15",
      "11S05",
      "11S20"
    ],
    "keywords": [
      "adic fields",
      "classifying extensions",
      "arbitrary prime degree",
      "cubic field extension",
      "finite extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}