Nieto, José H. On the divisibility of polynomials with integer coefficients. (Spanish. English summary) Zbl 1100.13506 Divulg. Mat. 11, No. 2, 149-152 (2003). Summary: Let \(f,g\in\mathbb{Z}[x]\). In this paper it is proved that \(g|f\) if and only if \(c(g)|c(f)\) (where \(c(f)\) denotes the content of \(f\), i.e., the greatest common divisor of its coefficients) and \(g(n)|f(n)\) for infinitely many \(n \in\mathbb{Z}\). As an application it is proved that the monic irreducible nonconstant polynomials \(f\in\mathbb{Z}[x]\) such that \(f(n)\) divides \(P(n^k)\) for all integers \(n\) \((k\geq 2\) being a fixed integer) are the cyclotomic polynomials \(\Phi_j\) with order \(j\) coprime with \(k\). Cited in 2 Documents MSC: 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 11C08 Polynomials in number theory PDFBibTeX XMLCite \textit{J. H. Nieto}, Divulg. Mat. 11, No. 2, 149--152 (2003; Zbl 1100.13506) Full Text: EuDML