# finite extensions of Dedekind domains are Dedekind

###### Theorem.

Let $R$ be a Dedekind domain with field of fractions $K$. If $L/K$ 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 $\mathcal{O}_{K}$. 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 FiniteExtensionsOfDedekindDomainsAreDedekind 2013-03-22 18:35:30 2013-03-22 18:35:30 gel (22282) gel (22282) 5 gel (22282) Theorem msc 13A15 msc 13F05 FiniteExtension DivisorTheoryInFiniteExtension