proof of finite separable extensions of Dedekind domains are Dedekind

Let R be a Dedekind domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K and L/K be a finite ( separable extensionMathworldPlanetmath of fields. We show that the integral closureMathworldPlanetmath A of R in L is also a Dedekind domain. That is, A is NoetherianPlanetmathPlanetmathPlanetmath (, integrally closedMathworldPlanetmath and every nonzero prime idealMathworldPlanetmathPlanetmath is maximal (

First, as integral closures are themselves integrally closed, A is integrally closed. Second, as integral closures in separable extensions are finitely generated, A is finitely generatedMathworldPlanetmathPlanetmath as an R-module. Then, any ideal 𝔞 of A is a submodule of A, so is finitely generated as an R-module and therefore as an A-module. So, A is Noetherian.

It only remains to show that a nonzero prime ideal 𝔭 of A is maximal. Choosing any p𝔭{0} there is a nonzero polynomialPlanetmathPlanetmath


for ckR, c00 and such that f(p)=0. Then


so 𝔭R is a nonzero prime ideal in R and is therefore a maximal idealMathworldPlanetmath. So,


gives an algebraic extensionMathworldPlanetmath of the field R/(𝔭R) to the integral domainMathworldPlanetmath A/𝔭. Therefore, A/𝔭 is a field (see a condition of algebraic extension) and 𝔭 is a maximal ideal.

Title proof of finite separable extensions of Dedekind domains are Dedekind
Canonical name ProofOfFiniteSeparableExtensionsOfDedekindDomainsAreDedekind
Date of creation 2013-03-22 18:35:36
Last modified on 2013-03-22 18:35:36
Owner gel (22282)
Last modified by gel (22282)
Numerical id 4
Author gel (22282)
Entry type Proof
Classification msc 13F05
Classification msc 13A15