If is a number field with ring of integers , then prove that if is a prime ideal of then is a prime ideal of .
It's easy to show that it's an ideal of but I'm struggling to show that it must be prime; any help would be very useful!
let and suppose that is the minimal polynomial of over it's a known fact that, since we have for all
now if then since we have and so if then because of minimality of thus gives us
and so
Note: in the proof we didn't need to be a prime ideal. so the above result holds for any non-zero ideal of