Irreducibility of a Polynomial Shifted by a Power of Another Polynomial

Abstract EN

In this note, we show that, for any f∈ℤx and any prime number p, there exists g∈ℤx for which the polynomial fx−gxp is irreducible over ℚ.

For composite p≥2, this assertion is not true in general.

However, it holds for any integer p≥2 if f is not of the form ahxk, where a≠0 and k≥2 are integers and h∈ℤx.