Determine whether $\sqrt{-5}$ is irreducible and/or prime in $\mathbb Z[\sqrt{-5}]$.
What is a prime $p>5$ which is prime in $\mathbb Z [\sqrt{-5}]$ not prime in $\mathbb Z [\sqrt{5}]$?
For the first one, I believe $\sqrt{−5}$ is irreducible as $N(\sqrt{−5}) = 5$ and the only integers that divide $5$ are $1$ (where all elements with norm $1$ are units) and $5$ (where only $±\sqrt{−5}$ have norm $5$). Is this a good explanation? I'm guessing that $\sqrt{-5}$ is prime but I'm not sure how to justify why.
For the second one, I can't think of any primes over 5.
Help would be great!