If $E/F$ is a finite extension and $E$ is algebraically closed, then $F$ is perfect.

Question

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.

Answer 1

You can prove:

• If every extension of F splits, then F is perfect.
• Let F ⊆ K ⊆ E a tower of fields. If E/F is algebraic and splits, then K/F splits.

And finally use the definition of algebraically closed.