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 $$K = \Q(\alpha)$$ for $$\alpha \in \mathcal O_K$$ and let $$f$$ be the minimal polynomial for $$\alpha$$ over $$\Z[x]$$. For any prime $$p$$ not dividing $$[\mathcal O_K : \Z[\alpha]]$$, 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 $$N(\mathfrak p_i) = p^{\deg \pi_i}$$.

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 $$\mathcal o = \Z, K=\Q, \mathcal O = \mathcal O_L$$).

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 $$\mathcal o[x]$$.