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On the divisibility of polynomials with integer coefficients. (Spanish. English summary) Zbl 1100.13506

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 \(k\).

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11C08 Polynomials in number theory
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