Geodesic
On the solvability of an Eisenstein trinomial of prime degree
Chahrazede Bouyacoub 1   ; Alain Salinier 2
1 Laboratoire d’Arithmétique, Codage, Combinatoire et Calcul formel LA3C, USTHB BP 32, El Alia, Bab Ezzouar 16111 Alger, Algeria
2 Pôle de Mathématiques et Informatique Laboratoire XLIM (UMR CNRS 7252) Université de Limoges 123, avenue Albert Thomas 87060 Limoges Cedex, France
Acta Arithmetica, Tome 178 (2017) no. 4, pp. 385-396 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $f(X)$.
DOI : 10.4064/aa8581-10-2016
Keywords: p p trinomial prime degree eisenstein respect where coprime rational integers investigate following question linked conjecture formulated lle schmid possible solvable nbsp mathbb main tool study determination decomposition group place above primes ramify splitting field

Chahrazede Bouyacoub  1   ; Alain Salinier  2

1 Laboratoire d’Arithmétique, Codage, Combinatoire et Calcul formel LA3C, USTHB BP 32, El Alia, Bab Ezzouar 16111 Alger, Algeria
2 Pôle de Mathématiques et Informatique Laboratoire XLIM (UMR CNRS 7252) Université de Limoges 123, avenue Albert Thomas 87060 Limoges Cedex, France 
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     title = {On the solvability of an {Eisenstein} trinomial of prime degree},
     journal = {Acta Arithmetica},
     pages = {385--396},
     year = {2017},
     volume = {178},
     number = {4},
     doi = {10.4064/aa8581-10-2016},
     language = {en},
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Chahrazede Bouyacoub; Alain Salinier. On the solvability of an Eisenstein trinomial of prime degree. Acta Arithmetica, Tome 178 (2017) no. 4, pp. 385-396. doi: 10.4064/aa8581-10-2016

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