---
date: 2022-01-25 10:10
modification date: Thursday 3rd February 2022 12:48:47
title: factorial
aliases: [factorial, "UFD"]

---

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# factorial
Every element admits a unique factorization into irreducibles, up to the action of units.

# Facts

- Every number field is a UFD
  
![](attachments/Pasted%20image%2020220203133117.png)
# Exercises

- Show PID $\implies$ UFD
![](attachments/Pasted%20image%2020220203130201.png)

- Show UFDs satisfy the ACC on principal ideals but are not necessarily Noetherian.
![](attachments/Pasted%20image%2020220203125908.png)

- Show irreducibles are prime.
![](attachments/Pasted%20image%2020220203125733.png)