{
  "id": "2304.12658",
  "version": "v1",
  "published": "2023-04-25T08:59:43.000Z",
  "updated": "2023-04-25T08:59:43.000Z",
  "title": "Galois groups of $\\binom{n}{0} + \\binom{n}{1} X + \\ldots + \\binom{n}{6} X^6$",
  "authors": [
    "Benjamin Klahn",
    "Marc Technau"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that the Galois group of the polynomial in the title is isomorphic to the full symmetric group on six symbols for all but finitely many $n$. This complements earlier work of Filaseta and Moy, who studied Galois groups of $\\binom{n}{0} + \\binom{n}{1} X + \\ldots + \\binom{n}{k} X^k$ for more general pairs $(n,k)$, but had to admit a possibly infinite exceptional set specifically for $k=6$ of at most logarithmic growth in $k$. The proof rests upon invoking Faltings' theorem on a suitable fibration of Galois resolvents to show that this exceptional set is, in fact, finite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-25T08:59:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R09",
      "11R32"
    ],
    "keywords": [
      "galois group",
      "complements earlier work",
      "full symmetric group",
      "possibly infinite exceptional set",
      "general pairs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}