Irreducibility of a Polynomial Shifted by a Power of Another Polynomial

Abstract

In this note, we show that, for any $f\in \mathbb{Z}\left[x\right]$ and any prime number $p$ , there exists $g\in \mathbb{Z}\left[x\right]$ for which the polynomial $f\left(x\right)-g{\left(x\right)}^p$ is irreducible over $\mathbb{Q}$ . For composite $p\ge 2$ , this assertion is not true in general. However, it holds for any integer $p\ge 2$ if $f$ is not of the form $ah{\left(x\right)}^k$ , where $a\ne 0$ and $k\ge 2$ are integers and $h\in \mathbb{Z}\left[x\right]$ .