$1+x^p+x^{2p}+\dotsb+x^{p(p-1)}$ irreducible

Question

Let $p$ be a prime number. Is $$f(x)=1+x^p+x^{2p}+\dotsb+x^{p(p-1)}$$ an irreducible polynomial over $\Bbb Z$?

Can we use Eisenstein's criterion?

$f(x+1)=1+(x+1)^p+(x+1)^{2p}+\dotsb+(x+1)^{p(p-1)}$

I am stuck. Thanks a lot!

Answer 1

The constant term of $f(x+1)$ is $p$. So you can use the Eisenstein's criterion for this question.