Irreducibility of $X^{p-1} + \cdots + X+1$ 
Can someone give me a hint how to the irreducibility of $X^{p-1} + \cdots + X+1$, where $p$ is a prime, in $\mathbb{Z}[X]$ ? 

Our professor gave us already one, namely to substitute $X$ with $X+1$, but I couldn't make much of that.

Hint: Let $y=x-1$. Note that our polynomial is $\dfrac{x^p-1}{x-1}$, which is
$\dfrac{(y+1)^p-1}{y}$. 
It is not difficult to show that $\binom{p}{k}$ is divisible by $p$ if $0\lt k\lt p$. Now use the Eisenstein Criterion.  
