Why $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible over $\mathbb{Q}$?

Question

I'm tring to know why $x^{p-1}+x^{p-2}+\cdots+1$ is irreducible over $\mathbb{Q}$.

Can you help me with a proof or showing me some references, please.

Answer 1

The usual trick is to look instead at $f(x+1) = \frac{(x+1)^p-1}{x}$. Since the binomial coefficients are divisible by $p$, this polynomial is Eisenstein at $p$, and therefore irreducible. Thus $f$ is also irreducible.