For which primes p is the polynomial irreducible in $\mathbb{Z}_p[x]$

Question

I'm studying for an upcoming exam, and have been given the following question:

For which primes p is the polynomial $f(x) = x^2 + x + 1$ irreducible in $\mathbb{Z}_p[x]$?

I think I can use Eisenstein's Criteria here, but I am confusing myself as I try to apply it. Any help is appreciated!

Answer 1

Hint:

For $p=2$ the polynomial is easily seen to be irreducible.

For odd prime $p$, the polynomial has roots in $\mathbb Z_p$ (and hence reducible) if and only if $4x^2+4x+4=(2x+1)^2+3$ has roots. Do you know under what circumstances $x^2\equiv-3\pmod p$ has integer solutions?