Reducibility of polynomials \(a_0(x)+a_1(x)y+a_2(x)y^2\) modulo \(p\)
type: Item
instance of: scholarly article
title: Reducibility of polynomials \(a_0(x)+a_1(x)y+a_2(x)y^2\) modulo \(p\)
zbMATH Open document ID: 0930.11009
author: Wolfgang M. Ruppert
published in: Archiv der Mathematik
publication date: 2000-02-16T00:00:00Z
review text: If \(f(x)=a_0(x)+a_1(x)y+a_2(x)y^2\in\mathbb{Z}[x,y]\) is an absolutely irreducible polynomial of degree \(m\) in \(x\), then the reduction of \(f\) modulo a given prime \(p\) will also be absolutely irreducible if \(p>c_mH(f)^{e_m}\) where \(H(f)\) is the height of \(f\), \(c_m\) is a specified constant depending on \(m\), and \(e_m=2m\) for \(m\geq 4\). The \(e_m\) are shown to be the best possible exponents if a certain conjecture of Bouniakowsky is assumed to be true.
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 12E05
zbMATH DE Number: 1292447
zbMATH Keywords: irreducible polynomial
zbMATH Keywords: reduction modulo a given prime
zbMATH Keywords: conjecture of Bouniakowsky
reviewed by: Richard A. Mollin
MaRDI profile type: Publication
full work available at URL: s000130050302
OpenAlex ID: W2033214097
DOI: 10.1007/S000130050302
description: scientific article; zbMATH DE number 1292447