---
aliases: "Dedekind"

---

A Dedekind domain is an integral domain that has the following three properties:
(i) [Noetherian](Noetherian.md),
(ii) [Integrally closed](Integrally%20closed.md),
(iii) All non-zero prime ideals are maximal.

![](attachments/Pasted%20image%2020220123194334.png)

![](attachments/Pasted%20image%2020220123194351.png)

![](attachments/Pasted%20image%2020220123201327.png)

Big list of equivalent characterizations:

![](attachments/Pasted%20image%2020220123203054.png)

- "To divide is to contain" for any commutative ring; this is an iff for Dedekind domains.
# Theorems/Results

![](attachments/Pasted%20image%2020220123202416.png)
![](attachments/Pasted%20image%2020220123202434.png)

## Exercises

- Show that if $R$ is Dedekind then all of its localizations at maximal ideals are DVRs.
- Use the fact that $R\da \CC[t]$ is a Dedekind domain to show that any complex polynomial has only finitely many roots.
- Prove that the integral closure of a Dedekind domain in a finite extension of its fraction field is also a Dedekind domain.
- Show that if $K$ is a number field then $\OO_K$ is a Dedekind domain.