Solution
= Solution
Define $\Phi_n(X)=\prod_{1\le k\le n,(k,n)=1}(X-e^{2\pi ik/n})$. From $X^n-1=\prod_{d\mid n}\Phi_d(X)$, induction and monic division show $\Phi_n\in\mathbb Z[X]$. Every $\Phi_n$ is irreducible over $\mathbb Q$. For prime $p$, $\Phi_p(X+1)=((X+1)^p-1)/X$ is Eisenstein at $p$, proving the assertion.