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. It is named after Richard Dedekind who developed the theorem based on the work of Ernst Kummer.

Statement for number fields

Let $K$ be a number field such that $K=\mathbb {Q} (\alpha )$ for $\alpha \in {\mathcal {O}}_{K}$ and let $f$ be the minimal polynomial for $\alpha$ over $\mathbb {Z} [x]$. For any prime $p$ not dividing $[{\mathcal {O}}_{K}:\mathbb {Z} [\alpha ]]$, write $$f(x)\equiv \pi _{1}(x)^{e_{1}}\cdots \pi _{g}(x)^{e_{g}}\mod p$$ where $\pi _{i}(x)$ are monic irreducible polynomials in $\mathbb {F} _{p}[x]$. Then $(p)=p{\mathcal {O}}_{K}$ factors into prime ideals as $$(p)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{g}^{e_{g}}$$ such that $N({\mathfrak {p}}_{i})=p^{\deg \pi _{i}}$, where $N$ is the ideal norm.

Statement for Dedekind domains

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

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