Unique factorization of ideals in a number field

Every nonzero proper ideal of a number ring factors uniquely as a finite product of positive powers of distinct nonzero prime ideals.

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 $p^{\mathrm{deg}{g}_i}$.

A number field $K$ is monogenic when its ring of integers has a power integral basis, meaning ${\mathcal{O}}_K=\mathbb{Z}[\theta ]$ for some algebraic integer $\theta$.

Let $K$ have degree $n$, and suppose a rational prime $p$ factors into $n$ distinct prime ideals of norm $p$. If $n>p$, then $K$ is not monogenic. Otherwise the Dedekind factorization theorem would force the degree-$n$ minimal polynomial of a power-basis generator to have $n$ distinct linear factors over ${\mathbb{F}}_p$, although only the $p$ polynomials $X-a$ are available.