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À propos d’une version faible du problème inverse de Galois

Volume 197 / 2021

Bruno Deschamps, François Legrand Acta Arithmetica 197 (2021), 55-76 MSC: Primary 12F10, 12F12, 20B25; Secondary 12E25, 12E30. DOI: 10.4064/aa190726-18-5 Published online: 24 September 2020

Abstract

This paper deals with the Weak Inverse Galois Problem, which, for a given field $k$, states that, for every finite group $G$, there exists a finite separable extension $L/k$ such that ${\rm {Aut}}(L/k)=G$. One of the paper’s goals is to explain how one can generically produce families of fields which fulfill this problem but which do not fulfill the usual Inverse Galois Problem. We show that this holds for, e.g., the fields $\mathbb {Q}^{{\rm sol}}$, $\mathbb {Q}^{{\rm tr}}$, $\mathbb {Q}^{{\rm pyth}}$, and for the maximal pro-$p$-extensions of $\mathbb {Q}$. Moreover, we show that, for every non-trivial finite group $G$, there exists a field fulfilling the Weak Inverse Galois Problem but over which $G$ does not occur as a Galois group. As a further application, we show that every field fulfills the regular version of the Weak Inverse Galois Problem.

Authors

  • Bruno DeschampsLaboratoire de Mathématiques Nicolas Oresme
    CNRS UMR 6139
    Université de Caen–Normandie
    BP 5186
    14032 Caen Cedex, France
    et
    Département de Mathématiques
    Le Mans Université
    Avenue Olivier Messiaen
    72085 Le Mans Cedex 9, France
    e-mail
  • François LegrandInstitut für Algebra
    Fachrichtung Mathematik
    TU Dresden
    01062 Dresden, Germany
    e-mail

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