Perfect field of characteristic $p>0$ which is not an algebraic extension of the prime field

Question

True/False

If $K$ is a perfect field of characteristic $p>0$, then is $K$ algebraic over $\mathbb{F}_p$?

My guess is no and I try to find a counterexample for this. Can anyone give me some hints?

Answer 1

Hint: algebraically closed fields are perfect.

Answer 2

Take $K=\mathbb F_p(X,X^{1/p},\dots,X^{1/p^n},\dots)$.