---
date: 2022-04-05 23:42
modification date: Wednesday 1st June 2022 16:43:38
title: "perfect field"
aliases: [perfect field, perfect]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# perfect field

A field $k$ is **perfect** iff 

- Every finite extension $L/k$ is automatically separable, or
- Either $k$ is characteristic zero, or $k$ is characteristic $p$ and the Frobenius $x\mapsto x^p$ is an automorphism.


# Examples

- Of perfect fields:
	- $\QQ, \RR, \CC$, any [[number field]]
	- Any finite field $\FF_q$
	- Any algebraically closed field
	- Any algebraic extension of a perfect field.
	- The perfect closure of $\FF_p(t)$, i.e. $\FF_p(t, t^{1\over p}, t^{1\over p^2}, \cdots)$.
- Of non-perfect fields:
	- $\FF_p(t)$, the rational function field in one variable over a finite field
	- $\FF_p\fls{t}$, the completion of $\FF_p(t)$.


![Pasted image 20210731183508](attachments/Pasted%20image%2020210731183508.png)