### Abstract

Let f(x) = (x - a_{1})...(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)^{2k} + 1 is irreducible for every k ≥ 1. In 1919 Pólya proved that if P(x) ∈ Z[x] is of degree m and there are m rational integer values a for which 0 < {pipe}P(a){pipe} < 2^{-N}N! where N = ⌉ m/2 ⌈, 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 {pipe}c{pipe} 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.

Original language | English |
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Pages (from-to) | 415-443 |

Number of pages | 29 |

Journal | Monatshefte fur Mathematik |

Volume | 163 |

Issue number | 4 |

DOIs | |

Publication status | Published - Aug 1 2011 |

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### Keywords

- Factors
- Irreducibility
- Polynomials
- Pólya-type
- Schur-type

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Monatshefte fur Mathematik*,

*163*(4), 415-443. https://doi.org/10.1007/s00605-010-0241-9