## 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.