Char $p$ fields, extension/subextension

Question

Let $K = \mathbb{F}_{p}$ (A prime field with char $= p$)

Let $C|K$ an algebraic closure of $K$

Then if $F$ is a field such that $K\subset F \subset C$, and $F \not= C \implies |F|<\infty$

Can someone help me?