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

• 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.

Examples

• Galois Field is Perfect
• Algebraically Closed Field is Perfect