Let $p$ be a prime number and let $n_1,n_2,\ldots,n_p$ be $p$ positive integers. Let $d=\gcd(n_1,n_2,\ldots,n_p)$. Show that the polynomial $$\frac{\left(\sum\limits_{i=1}^px^{n_i}\right)-p}{x^d-1}$$is irreducible in $\mathbb{Z}[X]$.
I'm not sure how to approach this question. Since $p$ is a prime then the question is begging me to use Eisenstein's Criterion but I can't figure out how to apply it. Should there be a clever substitution? Can someone point me in the right direction?