proof of Dedekind domains with finitely many primes are PIDs
Proof.
Let be all the primes of a Dedekind domain . If is any ideal of , then by the Weak Approximation Theorem we can choose such that for all (where is the -adic valuation
). But since is Dedekind, ideals have unique factorization
; since and have identical factorizations, we must have and is principal.∎