manuscripta mathematica

, Volume 99, Issue 2, pp 223–247

How often can a finite group be realized as a Galois group over a field?

  • C. U. Jensen
  • A. Prestel

DOI: 10.1007/s002290050171

Cite this article as:
Jensen, C. & Prestel, A. manuscripta math. (1999) 99: 223. doi:10.1007/s002290050171

Abstract:

Let G be any finite group and \(\) any class of fields. By \(\) we denote the minimal number of realizations of G as a Galois group over some field from the class \(\). For G abelian and \(\) the class of algebraic extensions of ℚ we give an explicit formula for \(\). Similarly we treat the case of an abelian p-group G and the class \(\) which is conjectured to be the class of all fields of characteristic ≠p for which the Galois group of the maximal p-extension is finitely generated. For non-abelian groups G we offer a variety of sporadic results.

Mathematics Subject Classification (1991): 12F10, 11R32 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • C. U. Jensen
    • 1
  • A. Prestel
    • 2
  1. 1.Matematisk Afdeling, Københavns Universitet, Universitetsparken 5,¶DK-2100 Copenhagen, Denmark. e-mail: cujensen@math.ku.dk}DK
  2. 2.Fakultät für Mathematik, Universität Konstanz, Postfach 5560, D-78434 Konstanz, Germany. e-mail: alex.prestel@uni-konstanz.deDE

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