Can there exist an algebraic integer ring with a prime ideal such that ?
Recall that if is prime, then it is maximal, and that is precisely . Since is maximal, is a field. We claim that no field of order 12 exists.
Suppose to the contrary that is a field of order 12. The characteristic of is then either 2 or 3, since it must be a prime dividing 12. Moreover, by Cauchy’s Theorem, contains nonzero elements and of (additive) order 2 and 3, respectively. That is, and , but no smaller multiple of or is 0. If has characteristic 2, then , and if has characteristic 3, then , both contradictions.
Since no field of order 12 exists, no prime ideal can have norm 12.