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On 2-extensions of the rationals with restricted ramification

Volume 163 / 2014

Peter Schmid Acta Arithmetica 163 (2014), 111-125 MSC: Primary 11F80, 11R32; Secondary 11S15. DOI: 10.4064/aa163-2-2

Abstract

For a finite group $G$ let ${\cal K}_2(G)$ denote the set of normal number fields (within ${\mathbb C}$) with Galois group $G$ which are $2$-ramified, that is, unramified outside $\{2,\infty \}$. We describe the $2$-groups $G$ for which ${\cal K}_2(G)\not =\varnothing $, and determine the fields in ${\cal K}_2(G)$ for certain distinguished $2$-groups $G$ appearing (dihedral, semidihedral, modular and semimodular groups). Our approach is based on Fröhlich's theory of central field extensions, and makes use of ring class field constructions (complex multiplication).

Authors

  • Peter SchmidMathematisches Institut
    Universität Tübingen
    Auf der Morgenstelle 10
    72076 Tübingen, Germany
    e-mail

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