Integer polynomials that are reducible modulo all primes. (English) Zbl 0603.12002

Brandl, Rolf

Am. Math. Mon. 93, 286-288 (1986).

In this note it is proved that for every positive integer \(n\neq 1\), which is not a prime, there exists a monic irreducible integer polynomial of degree \(n\) such that it is reducible modulo all primes. However, it is known that every monic irreducible polynomial of prime degree remains irreducible modulo infinitely many primes.

Reviewer: Nuria Vila (Barcelona)

MSC:

11C08	Polynomials in number theory
12E05	Polynomials in general fields (irreducibility, etc.)
11S20	Galois theory

Keywords:

irreducible integer polynomial; reducible modulo all primes