RO-TST 2010 Problem 19

Let $p$ be a prime number,let $n_1, n_2, \ldots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \ldots, n_p$ . Prove that the polynomial $$\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1}$$ is irreducible in $\mathbb{Q}[X]$ .

Beniamin Bogosel