For odd $n$ is the polynomial $X^{n-1}-X^{n-2}+...-X+1$ irreducible in $\mathbb{Q}[X]$?

Question

Let $n$ be an odd integer. Is the polynomial $X^{n-1}-X^{n-2}+...-X+1$ irreducible in $\mathbb{Q}[X]$?
I believe that it is, but I don't know how to ptove it. I tried to use Eisenstein's criterion, but it would only work when $n$ is prime. So, how to do it?

Answer 1

Let $n=9$. Then your polynomial $P_n$ factors (as mentioned in comments) as $\Phi_3(-x)\Phi_9(-x)$ where the $\Phi$ are cyclotomic polynomials. In general $P_n=\prod_{d\mid n,d>1}\Phi_d(-x)$, so it is irreducible precisely when $n$ is prime.

Answer 2

It's a (not too difficult) theorem that $d|n$ iff $(x^{d}-1)|(x^n-1)$. Factoring out $x-1$ and letting $n$ be odd should answer your question.

To prove the theorem, you only need the forward direction (which is the easier one). Just suppose $a$ is a root of $x^d -1$ and that $n= dk$. Then $a^d = a^{dk} = 1$, so $a$ is a root of $x^n - 1$.