 Dedekind factorization theorem

If ${\mathcal{O}}_K=\mathbb{Z}[\theta ]$ and the minimal polynomial of $\theta$ factors modulo a rational prime $p$ asthenand the prime ideal $(p,g_i(\theta ))$ has norm $p^{\mathrm{deg}{g}_i}$.

This gives an explicit instance of unique factorization of ideals in a number field.