A field extension, the larger field is algebraically closed, are there finite subextensions of arbitrary large degree?

Question

Let $L/F$ be a field extension of infinite degree. Assume that $L$ is algebraically closed. I am not assuming that $L$ is an algebraic closure of $F$.

Let $d\geq 1$. Must there be an intermediate field $F\subset K \subset L$ such that $K/F$ is finite and of degree at least $d$?

Answer 1

Let $L=\Bbb C$ and $F=\overline {\Bbb Q}$. Then an intermediate $K$ cannot be finite over $F$.