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.

Statement for number fields

Let $K$ be a number field such that $K=\mathbb{\mathbb{Q}}(\alpha )$ for $\alpha \in {{𝒪}}_K$ and let $f$ be the minimal polynomial for $\alpha$ over $\mathbb{\mathbb{Z}}[x]$. For any prime $p$ not dividing $[{{𝒪}}_K:\mathbb{\mathbb{Z}}[\alpha ]]$, write $$f(x)\equiv {\pi }_1(x{)}^{e_1}\cdots {\pi }_g(x{)}^{e_g}modp$$ where ${\pi }_i(x)$ are monic irreducible polynomials in ${\mathbb{𝔽}}_p[x]$. Then $(p)=p{{𝒪}}_K$ factors into prime ideals as $$(p)={\mathfrak{𝔭}}_1^{e_1}\cdots {\mathfrak{𝔭}}_g^{e_g}$$ such that $N({\mathfrak{𝔭}}_i)=p^{\mathrm{deg}{\pi }_i}$, 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 ${ℴ}$ 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 ${𝒪}$ the integral closure of ${ℴ}$. The above situation is just a special case as one can choose ${ℴ}=\mathbb{\mathbb{Z}},K=\mathbb{\mathbb{Q}},{𝒪}={{𝒪}}_L$). If $(0)\ne \mathfrak{𝔭}\subseteq {ℴ}$ is a prime ideal coprime to the conductor $\mathfrak{𝔉}=\{a\in {𝒪}\mid a{𝒪}\subseteq {ℴ}[\theta ]\}$ (i.e. their sum is ${𝒪}$). Consider the minimal polynomial $f\in {ℴ}[x]$ of $\theta$. The polynomial $\bar{f}\in ({ℴ}/\mathfrak{𝔭})[x]$ has the decomposition $$\bar{f}=\stackrel{e_1}{\overline{f_1}}\cdots \stackrel{e_r}{\overline{f_r}}$$ with pairwise distinct irreducible polynomials $\overline{f_i}$. The factorization of $\mathfrak{𝔭}$ into prime ideals over ${𝒪}$ is then given by $$\mathfrak{𝔭}={\mathfrak{𝔓}}_1^{e_1}\cdots {\mathfrak{𝔓}}_r^{e_r}$$ where ${\mathfrak{𝔓}}_i=\mathfrak{𝔭}{𝒪}+(f_i(\theta ){𝒪})$ and the $f_i$ are the polynomials $\overline{f_i}$ lifted to ${ℴ}[x]$.