Dedekind–Kummer theorem

Let ${\displaystyle K}$ be a number field such that ${\displaystyle K=\mathbb {Q} (\alpha )}$ for ${\displaystyle \alpha \in {\mathcal {O}}_{K}}$ and let ${\displaystyle f}$ be the minimal polynomial for ${\displaystyle \alpha }$ over ${\displaystyle \mathbb {Z} [x]}$. For any prime ${\displaystyle p}$ not dividing ${\displaystyle [{\mathcal {O}}_{K}:\mathbb {Z} [\alpha ]]}$, write

$${\displaystyle f(x)\equiv \pi _{1}(x)^{e_{1}}\cdots \pi _{g}(x)^{e_{g}}\mod p}$$

where ${\displaystyle \pi _{i}(x)}$ are monic irreducible polynomials in ${\displaystyle \mathbb {F} _{p}[x]}$. Then ${\displaystyle (p)=p{\mathcal {O}}_{K}}$ factors into prime ideals as

$${\displaystyle (p)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{g}^{e_{g}}}$$

such that ${\displaystyle N({\mathfrak {p}}_{i})=p^{\deg \pi _{i}}}$.[2]

See Neukirch.[1]