{
  "id": "2207.14113",
  "version": "v1",
  "published": "2022-07-28T14:27:34.000Z",
  "updated": "2022-07-28T14:27:34.000Z",
  "title": "On Galois groups of linearized polynomials related to the general linear group of prime degree",
  "authors": [
    "Rod Gow",
    "Gary McGuire"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\\mathbb{F}_q(t)$, where $t$ is transcendental over $\\mathbb{F}_q$. We prove that when $n$ is a prime, the Galois group is always $GL(n,q)$, except when $L(x)=x^{q^n}$. Equivalently, we prove that the arithmetic monodromy group of $L(x)/x$ is $GL(n,q)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-28T14:27:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general linear group",
      "galois group",
      "linearized polynomial",
      "prime degree"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}