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On the solvability of an Eisenstein trinomial of prime degree

Volume 178 / 2017

Chahrazede Bouyacoub, Alain Salinier Acta Arithmetica 178 (2017), 385-396 MSC: 11R32, 12F10. DOI: 10.4064/aa8581-10-2016 Published online: 26 April 2017

Abstract

Let $f(X) = X^p + a c^{p-2} X + a c^{p-1}$ be a trinomial of prime degree $p \ge 7$ that is Eisenstein with respect to $p$, where $a$ and $c$ are coprime rational integers. We investigate the following question, linked to a conjecture formulated by Kölle and Schmid: is it possible for $f(X)$ to be solvable over $\mathbb{Q}$? The main tool in this study is the determination of the decomposition group at a place above some primes that do not ramify in the splitting field of $f(X)$.

Authors

  • Chahrazede BouyacoubLaboratoire d’Arithmétique, Codage,
    Combinatoire et Calcul formel
    LA3C, USTHB
    BP 32, El Alia, Bab Ezzouar 16111
    Alger, Algeria
    e-mail
  • Alain SalinierPôle de Mathématiques et Informatique
    Laboratoire XLIM (UMR CNRS 7252)
    Université de Limoges
    123, avenue Albert Thomas
    87060 Limoges Cedex, France
    e-mail

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