caa a22 4500 445869429 CHVBK 20180317145507.0 cr unu---uuuuu 170323e20110801xx s 000 0 eng 10.1007/s00605-010-0241-9 doi (NATIONALLICENCE)springer-10.1007/s00605-010-0241-9 Irreducibility criteria of Schur-type and Pólya-type [Elektronische Daten] [K. Győry, L. Hajdu, R. Tijdeman] Let $${f(x)=(x-a_1)\cdots (x-a_m)}$$ , where a 1, . . . , a m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether $${(f(x))^{2^k}+1}$$ is irreducible for every k≥ 1. In 1919 Pólya proved that if $${P(x)\in\mathbb{Z}[x]}$$ is of degree m and there are m rational integer values a for which 0<|P(a)|<2−N N! where $${N=\lceil m/2\rceil}$$ , then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x)+c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation. The Author(s), 2010 Irreducibility nationallicence Factors nationallicence Polynomials nationallicence Schur-type nationallicence Pólya-type nationallicence Győry K. Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut Hajdu L. Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut Tijdeman R. Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RA, Leiden, The Netherlands aut Monatshefte für Mathematik Springer Vienna 163/4(2011-08-01), 415-443 0026-9255 163:4<415 2011 163 605 https://doi.org/10.1007/s00605-010-0241-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00605-010-0241-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Győry K. Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut NATIONALLICENCE 700 1- Hajdu L. Institute of Mathematics, University of Debrecen, P.O.Box 12, 4010, Debrecen, Hungary aut NATIONALLICENCE 700 1- Tijdeman R. Mathematical Institute, Leiden University, P.O.Box 9512, 2300 RA, Leiden, The Netherlands aut NATIONALLICENCE 773 0- Monatshefte für Mathematik Springer Vienna 163/4(2011-08-01), 415-443 0026-9255 163:4<415 2011 163 605 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer