# 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 ${\mathcal{O}}_{K}$. 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 |
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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 |