Let $k(x)$ be the rational function field of one variable over a field $k$. Are there infinitely many intermediate subfields of $k(x)/k$?
The motivation is as follows. Dedekind wrote in his supplement to Dirichlet's Lecture on Number Theory as follows.
Let $K$, $L$ be subfields of $\mathbb{C}$ such that $K \subset L$. If $L/K$ is a finite extension, it is easy to see by Galois theory that there are only finitely many intermidiate subfields of it. The converse is also true.
I tried to prove this. If $L/K$ is an infinite dimensional algebraic extension, it is easy to see that there are infinitely many intermidiate subfields. So the question is reduced to: Are there infinitely many intermidiate subfields if $L/K$ is not algebraic?
For any integer $k\gt 1$, consider the rational functions of $x^k$.