Number of prime ideals in Dedekind domain containing a given prime.

Question

I got stuck on the following problem:

Let $L/\mathbb{Q}$ be an extension of degree $4$ such that $\mathcal{O}_L=\mathbb{Z}[\alpha]$ is generated by a single element. Show that $\mathcal{O}_L$ has at most $3$ prime ideals that contain $3$.

I know that $\mathcal{O}_L$ is a Dedekind ring and thus the number of its prime ideals which contain $3$ is finite. Why it is at most $3$? Does it have something to do with the dimension of $L/\mathbb{Q}$?

Answer 1

By the Kummer-Dedekind theorem, if $3$ is contained in more than $3$ distinct prime ideals of $\Bbb{Z}[\alpha]$, then the minimal polynomial of $\alpha$ over $\Bbb{Z}$ splits into more than $3$ distinct irreducible factors over $\Bbb{F}_3$.