No prime ideal in an algebraic integer ring has norm 12

Can there exist an algebraic integer ring \mathcal{O} with a prime ideal A \subseteq \mathcal{O} such that N(A) = 12?

Recall that if A is prime, then it is maximal, and that N(A) is precisely |\mathcal{O}/A|. Since A is maximal, \mathcal{O} is a field. We claim that no field of order 12 exists.

Suppose to the contrary that F is a field of order 12. The characteristic of F is then either 2 or 3, since it must be a prime dividing 12. Moreover, by Cauchy’s Theorem, F contains nonzero elements \alpha and \beta of (additive) order 2 and 3, respectively. That is, 2\alpha = 0 and 3\beta = 0, but no smaller multiple of \alpha or \beta is 0. If F has characteristic 2, then \beta = 0, and if F has characteristic 3, then 4\alpha = \alpha = 0, both contradictions.

Since no field of order 12 exists, no prime ideal can have norm 12.

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