# proof of Dedekind domains with finitely many primes are PIDs

###### Proof.

Let ${\U0001d52d}_{1},\mathrm{\dots},{\U0001d52d}_{k}$ be all the primes of a Dedekind domain^{} $R$. If $I$ is any ideal of $R$, then by the Weak Approximation Theorem we can choose $x\in R$ such that ${\nu}_{{\U0001d52d}_{i}}((x))={\nu}_{{\U0001d52d}_{i}}(I)$ for all $i$ (where ${\nu}_{\U0001d52d}$ is the $\U0001d52d$-adic valuation^{}). But since $R$ is Dedekind, ideals have unique factorization^{}; since $(x)$ and $I$ have identical factorizations, we must have $(x)=I$ and $I$ is principal.
∎

Title | proof of Dedekind domains with finitely many primes are PIDs |
---|---|

Canonical name | ProofOfDedekindDomainsWithFinitelyManyPrimesArePIDs |

Date of creation | 2013-03-22 18:35:24 |

Last modified on | 2013-03-22 18:35:24 |

Owner | rm50 (10146) |

Last modified by | rm50 (10146) |

Numerical id | 4 |

Author | rm50 (10146) |

Entry type | Proof |

Classification | msc 13F05 |

Classification | msc 11R04 |