Dedekind–Kummer theorem

In algebraic number theory, the Dedekind–Kummer theorem describes how a prime ideal in a Dedekind domain factors over the domain's integral closure.^[1] It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer.

Let ${\displaystyle K}$ be a number field and ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠ the ring of algebraic integers in ⁠${\displaystyle K}$⁠. Let ${\displaystyle \alpha \in {\mathcal {O}}_{K}}$ and ${\displaystyle f}$ be the minimal polynomial of ${\displaystyle \alpha }$ over ${\displaystyle \mathbb {Z} [x]}$. For any prime ${\displaystyle p}$ not dividing the dimension ${\displaystyle [{\mathcal {O}}_{K}:\mathbb {Z} [\alpha ]]}$ of the free ⁠${\displaystyle \mathbb {Z} [\alpha ]}$⁠-module ⁠${\displaystyle {\mathcal {O}}_{K}}$⁠, 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, the ideal ${\displaystyle (p)=p{\mathcal {O}}_{K}}$ factors into prime ideals as $${\displaystyle (p)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{g}^{e_{g}}}$$ such that ${\displaystyle N({\mathfrak {p}}_{i})=p^{\deg \pi _{i}}}$, where ${\displaystyle N}$ is the ideal norm.^[2]

The Dedekind-Kummer theorem holds more generally than in the situation of number fields: Let ${\displaystyle {\mathcal {o}}}$ be a Dedekind domain contained in its quotient field ${\displaystyle K}$, ${\displaystyle L/K}$ a finite, separable field extension with ${\displaystyle L=K[\theta ]}$ for a suitable generator ${\displaystyle \theta }$ and ${\displaystyle {\mathcal {O}}}$ the integral closure of ${\displaystyle {\mathcal {o}}}$. The above situation is just a special case as one can choose ${\displaystyle {\mathcal {o}}=\mathbb {Z} ,K=\mathbb {Q} ,{\mathcal {O}}={\mathcal {O}}_{L}}$).

If ${\displaystyle (0)\neq {\mathfrak {p}}\subseteq {\mathcal {o}}}$ is a prime ideal coprime to the conductor ${\displaystyle {\mathfrak {F}}=\{a\in {\mathcal {O}}\mid a{\mathcal {O}}\subseteq {\mathcal {o}}[\theta ]\}}$ (i.e. their sum is ${\displaystyle {\mathcal {O}}}$). Consider the minimal polynomial ${\displaystyle f\in {\mathcal {o}}[x]}$ of ${\displaystyle \theta }$. The polynomial ${\displaystyle {\overline {f}}\in ({\mathcal {o}}/{\mathfrak {p}})[x]}$ has the decomposition $${\displaystyle {\overline {f}}={\overline {f_{1}}}^{e_{1}}\cdots {\overline {f_{r}}}^{e_{r}}}$$ with pairwise distinct irreducible polynomials ${\displaystyle {\overline {f_{i}}}}$. The factorization of ${\displaystyle {\mathfrak {p}}}$ into prime ideals over ${\displaystyle {\mathcal {O}}}$ is then given by $${\displaystyle {\mathfrak {p}}={\mathfrak {P}}_{1}^{e_{1}}\cdots {\mathfrak {P}}_{r}^{e_{r}}}$$ where ${\displaystyle {\mathfrak {P}}_{i}={\mathfrak {p}}{\mathcal {O}}+(f_{i}(\theta ){\mathcal {O}})}$ and the ${\displaystyle f_{i}}$ are the polynomials ${\displaystyle {\overline {f_{i}}}}$ lifted to ${\displaystyle {\mathcal {o}}[x]}$.^[1]