An irreducibility criterion for the sum of two relatively prime polynomials

Abstract

We extend a result of Cavachi on sums of relatively prime polynomials by proving that a polynomial of the form $f(X)+p^{k}g(X)$, with $f$ and $g$ relatively prime polynomials with integer coefficients, $\deg f<\deg g$, and $k$ a positive integer prime to $\deg g$ is irreducible over $\mathbb{Q}$ for all but finitely many prime numbers $p$.