Definition:Perfect Field/Definition 2

Definition

Let $F$ be a field.

$F$ 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

Examples

• Galois Field is Perfect
• Algebraically Closed Field is Perfect

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