Question 46
Let $p$ be a prime number, let $n_1, n_2, \dots, n_p$ be positive integer numbers, and let $d$ be the greatest common divisor of the numbers $n_1, n_2, \dots, n_p$. Prove that the polynomial
$$
\frac{X^{n_1} + X^{n_2} + \dots + X^{n_p} - p}{X^d - 1}
$$
is irreducible in $\mathbb{Q}[X]$.
$$
\frac{X^{n_1} + X^{n_2} + \dots + X^{n_p} - p}{X^d - 1}
$$
is irreducible in $\mathbb{Q}[X]$.