13A15-FiniteExtensionsOfDedekindDomainsAreDedekind.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{FiniteExtensionsOfDedekindDomainsAreDedekind}
\pmcreated{2013-03-22 18:35:30}
\pmmodified{2013-03-22 18:35:30}
\pmowner{gel}{22282}
\pmmodifier{gel}{22282}
\pmtitle{finite extensions of Dedekind domains are Dedekind}
\pmrecord{5}{41319}
\pmprivacy{1}
\pmauthor{gel}{22282}
\pmtype{Theorem}
\pmcomment{trigger rebuild}
\pmclassification{msc}{13A15}
\pmclassification{msc}{13F05}
%\pmkeywords{Dedekind domain}
%\pmkeywords{finite extension}
%\pmkeywords{integral closure}
\pmrelated{FiniteExtension}
\pmrelated{DivisorTheoryInFiniteExtension}

\endmetadata

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\begin{document}
\begin{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.
\end{theorem*}

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 \PMlinkname{unique factorization of ideals}{IdealDecompositionInDedekindDomain} is satisfied.

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