Dedekind–Kummer theorem

Short description: Theorem in algebraic number theory

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 [math]\displaystyle{ K }[/math] be a number field such that [math]\displaystyle{ K = \Q(\alpha) }[/math] for [math]\displaystyle{ \alpha \in \mathcal O_K }[/math] and let [math]\displaystyle{ f }[/math] be the minimal polynomial for [math]\displaystyle{ \alpha }[/math] over [math]\displaystyle{ \Z[x] }[/math]. For any prime [math]\displaystyle{ p }[/math] not dividing [math]\displaystyle{ [\mathcal O_K : \Z[\alpha]] }[/math], write[math]\displaystyle{ f(x) \equiv \pi_1 (x)^{e_1} \cdots \pi_g(x)^{e_g} \mod p }[/math]where [math]\displaystyle{ \pi_i (x) }[/math] are monic irreducible polynomials in [math]\displaystyle{ \mathbb F_p[x] }[/math]. Then [math]\displaystyle{ (p) = p \mathcal O_K }[/math] factors into prime ideals as[math]\displaystyle{ (p) = \mathfrak p_1^{e_1} \cdots \mathfrak p_g^{e_g} }[/math]such that [math]\displaystyle{ N(\mathfrak p_i) = p^{\deg \pi_i} }[/math].

Statement for Dedekind Domains

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

If [math]\displaystyle{ (0)\neq\mathfrak p\subseteq\mathcal o }[/math] is a prime ideal coprime to the conductor [math]\displaystyle{ \mathfrak F=\{a\in \mathcal O\mid a\mathcal O\subseteq\mathcal o[\theta]\} }[/math] (i.e. their product is [math]\displaystyle{ \mathcal O }[/math]). Consider the minimal polynomial [math]\displaystyle{ f\in \mathcal o[x] }[/math] of [math]\displaystyle{ \theta }[/math]. The polynomial [math]\displaystyle{ \overline f\in(\mathcal o / \mathfrak p)[x] }[/math] has the decomposition [math]\displaystyle{ \overline f=\overline{f_1}^{e_1}\cdots \overline{f_r}^{e_r} }[/math] with pairwise distinct irreducible polynomials [math]\displaystyle{ \overline{f_i} }[/math]. The factorization of [math]\displaystyle{ \mathfrak p }[/math] into prime ideals over [math]\displaystyle{ \mathcal O }[/math] is then given by [math]\displaystyle{ \mathfrak p=\mathfrak P_1^{e_1}\cdots \mathfrak P_r^{e_r} }[/math] where [math]\displaystyle{ \mathfrak P_i=\mathfrak p\mathcal O+(f_i(\theta)\mathcal O) }[/math] and the [math]\displaystyle{ f_i }[/math] are the polynomials [math]\displaystyle{ \overline{f_i} }[/math] lifted to [math]\displaystyle{ \mathcal o[x] }[/math].