finite extensions of Dedekind domains are Dedekind

Theorem.

Let $R$ be a Dedekind domain with field of fractions $K$. If $L\mathrm{/}K$ is a finite extension of fields and $A$ is the integral closure of $R$ in $L$, then $A$ is also a Dedekind domain.

For example, a number field $K$ is a finite extension of $\mathbb{Q}$ and its ring of integers is denoted by ${𝒪}_K$. Although such rings can fail to be unique factorization domains, the above theorem shows that they are always Dedekind domains and therefore unique factorization of ideals (http://planetmath.org/IdealDecompositionInDedekindDomain) 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