Polynomial Rings over fields with characteristic p

Let $\displaystyle char(\mathbb{F}) = p$, where $\displaystyle \mathbb{F}$ is a field. For what fields is $\displaystyle \frac{\mathbb{F}}{<x^2+7x+7>}$ an integral domain?

I tried using the Frobenius homomorphism, but I kept going in circles. Any ideas on how to proceed? (and please no direct Galois theory, if possible, as I don't know any)

Re: Polynomial Rings over fields with characteristic p

F should be a field over which the polynomial is irreducible.