# 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 ProofOfDedekindDomainsWithFinitelyManyPrimesArePIDs 2013-03-22 18:35:24 2013-03-22 18:35:24 rm50 (10146) rm50 (10146) 4 rm50 (10146) Proof msc 13F05 msc 11R04