The problem is
Let $E$ be a finite extension of $F$ and suppose $E$ is algebraically closed. Show that $F$ is perfect.
I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is separable, and a field $F$ is perfect if and only if it has characteristic $0$, or it has characteristic $p$ and $F=F^p$; then, fields of characteristic $0$ and finite fields are perfect.