Definition:Perfect Field

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Definition

Let $F$ be a field.


Definition 1

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


Definition 2

The field $F$ is perfect if and only if 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


Examples