proof of Dedekind domains with finitely many primes are PIDs

Proof.

Let $\mathfrak{p}_{1},\ldots,\mathfrak{p}_{k}$ 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 $\nu_{\mathfrak{p}_{i}}((x))=\nu_{\mathfrak{p}_{i}}(I)$ for all $i$ (where $\nu_{\mathfrak{p}}$ is the $\mathfrak{p}$ -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