{
  "id": "2209.10638",
  "version": "v1",
  "published": "2022-09-21T20:09:21.000Z",
  "updated": "2022-09-21T20:09:21.000Z",
  "title": "On the shapes of pure prime degree number fields",
  "authors": [
    "Erik Holmes"
  ],
  "comment": "36 pages, 1 figure. Comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "For $p$ prime and $\\ell = \\frac{p-1}{2}$, we show that the shapes of pure prime degree number fields lie on one of two $\\ell$-dimensional subspaces of the space of shapes, and which of the two subspaces is dictated by whether or not $p$ ramifies wildly. When the fields are ordered by absolute discriminant we show that the shapes are equidistributed, in a regularized sense, on these subspaces. We also show that the shape is a complete invariant within the family of pure prime degree fields. This extends the results of Harron, in [Har17], who studied shapes in the case of pure cubic number fields. Furthermore we translate the statements of pure prime degree number fields to statements about Frobenius number fields, $F_p = C_p\\rtimes C_{p-1}$, with a fixed resolvent field. Specifically we show that this study is equivalent to the study of $F_p$-number fields with fixed resolvent field $\\mathbb{Q}(\\zeta_p)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-21T20:09:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R21",
      "11R45",
      "11E12",
      "11P21"
    ],
    "keywords": [
      "pure prime degree number fields",
      "prime degree number fields lie",
      "fixed resolvent field",
      "pure cubic number fields",
      "pure prime degree fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}