proof of Dedekind domains with finitely many primes are PIDs
Proof.
Let 𝔭1,…,𝔭k 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∈R such that ν𝔭i((x))=ν𝔭i(I) for all i (where ν𝔭 is the 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 |