An irreducibility criterion for the sum of two relatively prime polynomials (Q6548055)

The authors study the following problem. Given two non-zero relatively prime polynomials \(f(x),g(x)\) with \(\deg f<\deg g\) in \({\mathbb{Z}}[x]\), find an infinite set \({\mathcal{N}}\subseteq {\mathbb{N}}\) such that \(f(x)+Ng(x)\) is irreducible in \({\mathbb{Q}}\) for \(N\in {\mathcal{N}}\).\par \textit{N. C. Bonciocat} et al. [Int. J. Number Theory 9, No. 6, 1529--1539 (2013; Zbl 1303.11119)] prove that if \({\mathcal{N}}\) is the set of prime powers \(p^{\alpha}\), where \(\alpha\) is a positive integer such that \(\gcd(\alpha, \deg g -\deg f)=1\) and \(p\) is a prime number that divides none of the leading coefficients of \(f(x)\) and \(g(x)\), then the polynomial \(f(x)+Ng(x)\) is irreducible over \({\mathbb{Q}}\) for all but finitely many positive integers \(N\) in \({\mathcal{N}}\).\par The main result of this paper, Theorem 3.3, extends the Bonciocat et al. result, and implies that \(f(x)+Ng(x)\) is irreducible over \({\mathbb{Q}}\) for all but finitely many square-free positive integers \(N\), where \(f(x),g(x)\in{\mathbb{Z}}[x]\) are non-zero relatively prime polynomials with \(\deg f<\frac 12 \deg g\). Furthermore, it is shown that the condition \(\deg f<\frac 12 \deg g\) is sharp.\par The second main result, Theorem 4.1, is devoted to determining all polynomials \(g(x)\in{\mathbb{Z}}[x]\) and \(r\in{\mathbb{Z}}\) such that \(r+p^2g(x)\) is irreducible over \({\mathbb{Q}}\) for a sufficiently large prime number \(p\).