Irreducible polynomials which are locally reducible everywhere

Abstract

For any positive integer n n , there exist polynomials f ( x ) ∈ Z [ x ] f(x)\in \mathbb {Z}[x] of degree n n which are irreducible over Q \mathbb {Q} and reducible over Q p \mathbb {Q}_{p} for all primes p p if and only if n n is composite. In fact, this result holds over arbitrary global fields.