# Definition:Perfect Field

Jump to navigation
Jump to search

## Definition

Let $F$ be a field.

### Definition 1

The field $F$ is **perfect** if and only if $F$ has no inseparable extensions.

### Definition 2

The field $F$ is **perfect** if and only if one of the following holds:

- $\operatorname{char}(F) = 0$
- $\operatorname{char}(F) = p$ with $p$ prime and $\operatorname{Frob}$ is an automorphism of $F$

where:

- $\operatorname{char}(F)$ denotes the characteristic of $F$
- $\operatorname{Frob}$ denotes the Frobenius endomorphism on $F$

## Also see

- Equivalence of Definitions of Perfect Field
- Frobenius Endomorphism on Field is Injective
- Definition:Separably Closed Field, a field with no separable extensions