Algebraically Closed Field is Perfect

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.

$\blacksquare$