Definition:Perfect Field

Definition
Let $F$ be a field.

Definition 1
The field $F$ is perfect $F$ has no inseparable extensions.

Definition 2
The field $F$ is perfect 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

Examples

 * Finite Field is Perfect
 * Algebraically Closed Field is Perfect