{
  "id": "math/0308069",
  "version": "v2",
  "published": "2003-08-07T16:49:14.000Z",
  "updated": "2004-05-04T13:21:01.000Z",
  "title": "The conjugate dimension of algebraic numbers",
  "authors": [
    "Neil Berry",
    "Arturas Dubickas",
    "Noam D. Elkies",
    "Bjorn Poonen",
    "Chris Smyth"
  ],
  "comment": "17 pages",
  "journal": "Quart. J. Math. 55 (2004), 237-252",
  "doi": "10.1093/qmath/hah003",
  "categories": [
    "math.NT"
  ],
  "abstract": "We find sharp upper and lower bounds for the degree of an algebraic number in terms of the $Q$-dimension of the space spanned by its conjugates. For all but seven nonnegative integers $n$ the largest degree of an algebraic number whose conjugates span a vector space of dimension $n$ is equal to $2^n n!$. The proof, which covers also the seven exceptional cases, uses a result of Feit on the maximal order of finite subgroups of $GL_n(Q)$; this result depends on the classification of finite simple groups. In particular, we construct an algebraic number of degree 1152 whose conjugates span a vector space of dimension only 4. We extend our results in two directions. We consider the problem when $Q$ is replaced by an arbitrary field, and prove some general results. In particular, we again obtain sharp bounds when the ground field is a finite field, or a cyclotomic extension of $Q$. Also, we look at a multiplicative version of the problem by considering the analogous rank problem for the multiplicative group generated by the conjugates of an algebraic number.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-05-04T13:21:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R06"
    ],
    "keywords": [
      "algebraic number",
      "conjugate dimension",
      "conjugates span",
      "vector space",
      "seven exceptional cases"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......8069B"
    }
  }
}