Dedekind–Kummer theorem

Template:Short description 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.Template:CnTemplate:When

Statement for number fields

Let $K$ be a number field and Template:Tmath the ring of algebraic integers in Template:Tmath. Let $\alpha \in {𝒪}_K$ be such that $\mathbb{Q}[\alpha ]=K$ and $f$ be the minimal polynomial of $\alpha$ over $\mathbb{Z}[x]$. For any prime $p$ not dividing the index $[{𝒪}_K:\mathbb{Z}[\alpha ]]$, 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 ${𝔽}_p[x]$. Then, the ideal $(p)=p{𝒪}_K$ factors into prime ideals as $$(p)={𝔭}_1^{e_1}\cdots {𝔭}_g^{e_g}$$ such that $N({𝔭}_i)=p^{\mathrm{deg}{\pi }_i}$, 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 $ℴ=\mathbb{Z},K=\mathbb{Q},𝒪={𝒪}_L$).

If $(0)\ne 𝔭\subseteq ℴ$ is a prime ideal coprime to the conductor $𝔉=\{a\in 𝒪\mid a𝒪\subseteq ℴ[\theta ]\}$ (i.e. their sum is $𝒪$). Consider the minimal polynomial $f\in ℴ[x]$ of $\theta$. The polynomial $\bar{f}\in (ℴ/𝔭)[x]$ has the decomposition $$\bar{f}={\overline{f_1}}^{e_1}\cdots {\overline{f_r}}^{e_r}$$ with pairwise distinct irreducible polynomials $\overline{f_i}$. The factorization of $𝔭$ into prime ideals over $𝒪$ is then given by $$𝔭={𝔓}_1^{e_1}\cdots {𝔓}_r^{e_r}$$ where ${𝔓}_i=𝔭𝒪+(f_i(\theta )𝒪)$ and the $f_i$ are the polynomials $\overline{f_i}$ lifted to $ℴ[x]$.