Proving $1+x+\cdots +x^{p-2}+x^{p-1}$ is irreducible for prime $p$ [duplicate]

Prove that if $p$ is a prime number, the polynomial $f(x)=1+x+\cdots +x^{p-2}+x^{p-1}$ is irreducible in $\mathbb{Z}[x]$.

I tried using Eisenstein Criterion and the Rational Root Theorem, also, I know that $f$ is irreducible over $\mathbb{Z}$ if and only if it is irreducible over $\mathbb{Q}$ because of the Gauss' Lemma. Maybe writing $x^p-1=(x-1)f(x)$ and proving $x^p-1$ dos not have any other roots may help, but I don't know.

You are really close, from $f(x)=\dfrac{x^p-1}{x-1}$ we get using the Binomial Theorem that$$f(x+1)=x^{p-1}+\sum \limits _{k=1}^{p-1}\binom{p}{k}x^{k-1}.$$Using that $p\mid \binom{p}{k}$ for $1\leq k\leq p-1$ and Eisenstein Criterion you get that $f(x+1)$ is irreducible over $\mathbb{Q}$, from Gauss' Lemma follows that $f(x)$ is irreducible over $\mathbb{Z}$.