On the Factorization of lacunary polynomials

This paper addresses the factorization of polynomials of the form $F(x)=f_0(x)+f_1(x)x^n+\cdots +f_{r-1}(x)x^{(r-1)n}+f_r(x)x^{rn}$ where $r$ is a fixed positive integer and the $f_j(x)$ are fixed polynomials in $\mathbb{Z}[x]$ for $0\le j\le r$. We provide an efficient method for showing that for $n$ sufficiently large and reasonable conditions on the $f_j(x)$, the non-reciprocal part of $F(x)$ is either $1$ or irreducible. We illustrate the approach including giving two examples that arise from trace fields of hyperbolic $3$-manifolds.