I tried this problem for a while, but didn't see the application of Eisenstein's irreducibility criterion here. All the coefficients, including the leading coefficient, are equal to 1.
p is a prime number.
The polynomial is $f(x) = x^{p-1} + x^{p-2} + \cdots + x + 1$
How do I proceed? Many thanks.
$f(x) = \frac{x^p - 1}{x-1}$. [This division is valid in the fraction field of polynomials.]
Thus $f(t+1) = \frac{(t+1)^p - 1}{t}$.
Eisenstein's criterion applies and so $f(t+1)$ is irreducible in $\mathbb{Z}[t]$.
Now $f(x)$ cannot be reducible in $\mathbb{Z}[x]$ otherwise you could substitute $x$ by $(t+1)$ in a non-trivial factorization of $f(x)$ to get a non-trivial factorization of $f(t+1)$.