Galois group of \(X^ p+ aX+a\)
type: Item
instance of: scholarly article
title: Galois group of \(X^ p+ aX+a\)
zbMATH Open document ID: 0863.12003
published in: Journal of Algebra
publication date: 1996-11-10T00:00:00Z
review text: In this paper the Galois group \(G\) over \(\mathbb{Q}\) of trinomials of the form \(f(X)= X^p+ aX+ a\) is studied, where \(p\) is a prime number and \(a\) is an integer such that \(p\) divides \(a\) exactly once (Eisenstein type trinomial with respect to \(p)\). The author determines the inertia group of \(p\) in \(N/ \mathbb{Q}\) and the splitting field of \(f(X)\), and he proves that the Galois group \(G\) is the symmetric group \(S_p\) or the group \(\text{Aff} (\mathbb{F}_p)\) of all affine transformations over \(\mathbb{F}_p\). Furthermore, he shows that \(G \simeq S_p\) in each of the two following cases: if \(a<0\) or if \(a/p \not\equiv 1 \pmod p\).
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 870176
zbMATH Keywords: Eisenstein polynomial
zbMATH Keywords: Galois group
author: A. Movahhedi
reviewed by: Nuria Vila
MaRDI profile type: Publication
full work available at URL: jabr.1996.0104
OpenAlex ID: W2054998825
DOI: 10.1006/JABR.1996.0104
description: scientific article; zbMATH DE number 870176