How to prove that this polynomial is irreducible?

Question

Let p be a prime. How to prove that $x^{p-1}+x^{p-2}+...+1$ is irreducible over $Q[x]$?

Answer 1

Denote by $f(x)$ this polynomial. Recall that $f(x)$ is reducible iff $f(x+1)$ is reducible.

$$f(x) = \frac{x^p - 1}{x - 1}\\ \therefore f(x+1) = \frac1{x} ((x+1)^p -1) = \frac1{x} \left( \sum_{t=1}^p {p \choose t} x^t \right) = \sum_{t=1}^p {p \choose t} x^{t-1}$$

Now use the Eisenstein criterion.