I'm not quite sure what you're asking exactly, but a prime number in a quadratic field will always have a norm that is prime or the prime squared.
I'm aware that if p is an integer in a Q field, and if its norm is prime, then p is a prime in Q. However, what if I myself need to confirm whether or not a number is prime in a specific field? Is there anyway I can identify/confirm that a number is prime when its properties are exclusive of this theorem?