A family of tests for irreducibility of polynomials (Q2809188)

In this article, the following irreducibility result is proved. Let \(f(x) = a_nx^n + \cdots + a_0\) be a polynomial with integral coefficients and suppose that there is a prime number \(p\) with the following properties: a) \(p\) does not divide \(a_n\); b) \(p\) divides \(a_j\) for \(0 \leq j \leq n-1\); c) there exists an index \(k\) with \(1 \leq k \leq n-1\) such that \(p^2\) does not divide \(a_k\). Let \(m\) denote the smallest such index. Assume moreover that there is a prime number \(q \neq p\) such that the reduction of \(f\) modulo \(q\) does not have an irreducible factor of degree \(\ell\) for \(1 \leq \ell \leq m\). Then \(f\) is irreducible in \({\mathbb Z}[x]\).NEWLINENEWLINEThe proof is an immediate consequence of the results obtained in [\textit{S. H. Weintraub}, Proc. Am. Math. Soc. 141, No. 4, 1159--1160 (2013; Zbl 1271.12001)].