# Algebraically Closed Field is Perfect

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## Theorem

Let $F$ be an algebraically closed field.

Then $F$ is perfect.

## Proof

Let $E / F$ be any algebraic extension.

Since $F$ is an algebraically closed field, $E = F$.

By Field is Separable over itself, $E$ is separable over $F$.

Hence $F$ is perfect.

$\blacksquare$