If x is any odd standard rational integer what can be said about the greatest common factor of the quadratic integers $(x-\sqrt{-2})$ and $(x+\sqrt{-2})$ in $\Bbb{Q}$$[\sqrt{-2}]$?
I know that these numbers can be prime if the norm is a prime number, but other than that I'm completely lost.
Note that in $\Bbb{Z}[\sqrt{-2}]$, Which is a UFD, the prime decomposition of $2$ is $$2= (\sqrt{-2})(-\sqrt{-2})$$
And this is because as you said, $\pm \sqrt{-2}$ have prime norm and hence are prime.
So if a prime = irreducible element $p$ divides $2x$ and $2\sqrt{-2}$ then it has to divide $2$ or $\sqrt{-2}$ which itself is prime. Then, $p=\pm \sqrt{-2}$. But: $$\frac{x+\sqrt{-2}}{\pm \sqrt{-2}}=\mp\frac{-2+x\sqrt{-2}}{2}=\pm 1 \mp \frac{x\sqrt{-2}}{2}\not \in \Bbb{Z}[\sqrt{-2}]$$
As $x$ is odd