{
  "id": "2408.12441",
  "version": "v1",
  "published": "2024-08-22T14:37:09.000Z",
  "updated": "2024-08-22T14:37:09.000Z",
  "title": "Automorphism Groups of Finite Extensions of Fields and the Minimal Ramification Problem",
  "authors": [
    "Alexei Entin"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the following question: given a global field $F$ and finite group $G$, what is the minimal $r$ such that there exists a finite extension $K/F$ with $\\mathrm{Aut}(K/F)\\cong G$ that is ramified over exactly $r$ places of $F$? We conjecture that the answer is $\\le 1$ for any global field $F$ and finite group $G$. In the case when $F$ is a number field we show that the answer is always $\\le 4[F:\\mathbb Q]$. In the case $F=\\mathbb Q$ we show that the answer equals 1 (for nontrivial $G$) if one assumes Schinzel's Hypothesis H. We show that the answer is always $\\le 1$ if $F$ is a global function field. We also show that for a broader class of fields $F$ than previously known, every finite group $G$ can be realized as the automorphism group of a finite extension $K/F$ (without restriction on the ramification). An important new tool used in this work is a recent result of the author and C. Tsang, which says that for any finite group $G$ there exists a natural number $n$ and a subgroup $H\\leqslant S_n$ of the symmetric group such that $N_{S_n}(H)/H\\cong G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-22T14:37:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite extension",
      "minimal ramification problem",
      "automorphism group",
      "finite group",
      "global field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}