Английская Википедия:Dedekind–Kummer theorem

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Шаблон: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.

Statement for number fields

Let <math>K </math> be a number field such that <math>K = \Q(\alpha)</math> for <math>\alpha \in \mathcal O_K</math> and let <math>f</math> be the minimal polynomial for <math>\alpha</math> over <math>\Z[x]</math>. For any prime <math>p</math> not dividing <math>[\mathcal O_K : \Z[\alpha]]</math>, write<math display="block">f(x) \equiv \pi_1 (x)^{e_1} \cdots \pi_g(x)^{e_g} \mod p</math>where <math>\pi_i (x)</math> are monic irreducible polynomials in <math>\mathbb F_p[x]</math>. Then <math>(p) = p \mathcal O_K</math> factors into prime ideals as<math display="block">(p) = \mathfrak p_1^{e_1} \cdots \mathfrak p_g^{e_g}</math>such that <math>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>\mathcal o</math> be a Dedekind domain contained in its quotient field <math>K</math>, <math>L/K</math> a finite, separable field extension with <math>L=K[\theta]</math> for a suitable generator <math>\theta</math> and <math>\mathcal O</math> the integral closure of <math>\mathcal o</math>. The above situation is just a special case as one can choose <math>\mathcal o = \Z, K=\Q, \mathcal O = \mathcal O_L</math>).

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