{
  "id": "0709.2868",
  "version": "v1",
  "published": "2007-09-18T16:14:46.000Z",
  "updated": "2007-09-18T16:14:46.000Z",
  "title": "On Galois Groups of Prime Degree Polynomials with Complex Roots",
  "authors": [
    "Oz Ben-Shimol"
  ],
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "Let $f$ be an irreducible polynomial of prime degree $p\\geq 5$ over $\\QQ$, with precisely $k$ pairs of complex roots. Using a result of Jens H\\\"{o}chsmann (1999), we show that if $p\\geq 4k+1$ then $\\Gal(f/\\QQ)$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T.Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\\QQ$ having complex roots.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-09-18T16:14:46.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "galois group",
      "prime degree polynomials",
      "complex roots",
      "irreducible polynomial",
      "frobenius group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.2868B"
    }
  }
}