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.

单词	FiniteExtensionsOfDedekindDomainsAreDedekind
释义	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.
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