finite extensions of Dedekind domains are Dedekind


Let R be a Dedekind domainMathworldPlanetmath with field of fractionsMathworldPlanetmath K. If L/K is a finite extensionMathworldPlanetmath of fields and A is the integral closureMathworldPlanetmath of R in L, then A is also a Dedekind domain.

For example, a number fieldMathworldPlanetmath K is a finite extension of and its ring of integers is denoted by 𝒪K. Although such rings can fail to be unique factorization domainsMathworldPlanetmath, the above theorem shows that they are always Dedekind domains and therefore unique factorization of ideals ( is satisfied.

Title finite extensions of Dedekind domains are Dedekind
Canonical name FiniteExtensionsOfDedekindDomainsAreDedekind
Date of creation 2013-03-22 18:35:30
Last modified on 2013-03-22 18:35:30
Owner gel (22282)
Last modified by gel (22282)
Numerical id 5
Author gel (22282)
Entry type Theorem
Classification msc 13A15
Classification msc 13F05
Related topic FiniteExtension
Related topic DivisorTheoryInFiniteExtension