On the solvability of an Eisenstein trinomial of prime degree
Volume 178 / 2017
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 $f(X)$.
