# Definition:Perfect Field

## 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

• Results about perfect fields can be found here.