For what primes $p$ is the mod 2 polynomial $(x^p-1)/(x-1)$ irreducible? Are there simple criteria?

Question

The polynomial $(x^p-1)/(x-1)$ is irreducible over the integers exactly when $p$ is a prime. But it is only sometimes irreducible when considered as a polynomial over $Z_2$. Are there any simple criteria for $p$ to tell when that happens? What about for other prime moduli besides 2?