{
  "id": "2006.11870",
  "version": "v1",
  "published": "2020-06-21T18:35:06.000Z",
  "updated": "2020-06-21T18:35:06.000Z",
  "title": "Genus fields of Kummer $\\ell^n$-cyclic extensions",
  "authors": [
    "Carlos Daniel Reyes-Morales",
    "Gabriel Villa-Salvador"
  ],
  "comment": "19 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a construction of the genus field for Kummer $\\ell^n$-cyclic extensions of rational congruence function fields, where $\\ell$ is a prime number. First, we compute the genus field of a field contained in a cyclotomic function field, and then for the general case. This generalizes the result obtained by Peng for a Kummer $\\ell$-cyclic extension. Finally, we study the extension $(K_1K_2)_{\\frak{ge}}/(K_1)_{\\frak{ge}}(K_2)_{\\frak{ge}}$, for $K_1$, $K_2$ abelian extensions of $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-21T18:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R60",
      "11R29",
      "11R58"
    ],
    "keywords": [
      "genus field",
      "cyclic extension",
      "rational congruence function fields",
      "cyclotomic function field",
      "abelian extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}