{
  "id": "math/0612528",
  "version": "v4",
  "published": "2006-12-18T20:03:30.000Z",
  "updated": "2007-01-28T15:38:33.000Z",
  "title": "Polynomials with roots in ${\\Bbb Q}_p$ for all $p$",
  "authors": [
    "Jack Sonn"
  ],
  "comment": "6 pages, revised to simplify a proof, improve a result, add a remark, and make some minor corrections",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $f(x)$ be a monic polynomial in $\\dZ[x]$ with no rational roots but with roots in $\\dQ_p$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $f(x)$ is a product of $m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $m$ proper subgroups. We prove that for any $m>1$, every finite solvable group which is a union of conjugates of $m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with $m=2$) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric--i.e. regular-- extension of $\\dQ(t)$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2007-01-28T15:38:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R32",
      "12F12"
    ],
    "keywords": [
      "galois group",
      "proper subgroups",
      "irreducible polynomials",
      "conjugates",
      "finite solvable group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math.....12528S"
    }
  }
}