Problem related to Cyclotomic Polynomials

Question

I'm trying to show that if $p$ is prime, then $$x^{p-1}-x^{p-2}+x^{p-3}-...-x+1$$ is irreducible over $\mathbb{Q}$.

I don't have an idea of how to start. I know the $p^{th}$ cyclotomic polynomial is irreducible.

Answer 1

The canonical way to do this starts as follows: Rewrite the polynomial as $\frac{x^p+1}{x+1}$ and substitute $x-1$ for $x$. See if you can get anywhere with that.

If you give up, a similar result is proved here