# Definition:Perfect Field

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## Definition

Let $F$ be a field.

### Definition 1

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

### Definition 2

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

- $\Char F = 0$
- $\Char F = p$ with $p$ prime and $\Frob$ is an automorphism of $F$

where:

- $\Char F$ denotes the characteristic of $F$
- $\Frob$ 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

- Results about
**perfect fields**can be found here.