{
  "id": "0812.2167",
  "version": "v1",
  "published": "2008-12-11T14:58:35.000Z",
  "updated": "2008-12-11T14:58:35.000Z",
  "title": "Explicit Constructions of the non-Abelian $\\mathbf{p^3}$-Extensions Over $\\mathbf{\\QQ}$",
  "authors": [
    "Oz Ben-Shimol"
  ],
  "comment": "12 pages. keywords: Constructive Galois Theory, Heisenberg group, Explicit Embedding problem, Minimal Ramification",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime. Let $F/k$ be a cyclic extension of degree $p$ and of characteristic different from $p$. The explicit constructions of the non-abelian $p^{3}$-extensions over $k$, are induced by certain elements in ${F(\\mu_{p})}^{*}$. In this paper we let $k=\\QQ$ and present sufficient conditions for these elements to be suitable for the constructions. Polynomials for the non-abelian groups of order 27 over $\\QQ$ are constructed. We describe explicit realizations of those groups with exactly two ramified primes, without consider Scholz conditions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-12-11T14:58:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12F12",
      "11R18"
    ],
    "keywords": [
      "explicit constructions",
      "odd prime",
      "cyclic extension",
      "scholz conditions",
      "non-abelian groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0812.2167B"
    }
  }
}