I'm trying to determine if $(7)$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$ so what I did was:
$\mathbb{Z}[\sqrt{-5}]/ (7) \cong \mathbb{Z}[X]/(x^2+5,7) \cong \mathbb{F}_7[X]/(x^2+5) \cong \mathbb{F}_7[X]/(x^2-2) $
Now I'm pretty sure that $x^2+5$ and $x^2-2$ and prime in this field, so the quotient field is an integral domain. So then $(7)$ is prime in $\mathbb{Z}[\sqrt{-5}]$. But I read somewhere that $(7)$ is not a prime ideal so I don't know where I went wrong? Am I wrong to say that the polynomial $x^2+5$ or $x^2-2$ is prime?