Find a finite extension of $\mathbb F_2(x,y)$ having infinitely many intermediate subfields.

Question

Let $\mathbb F_2$ be a field of order $2$. Let $F$ be the function field $\mathbb F_2(x,y)$, i.e., $x,y$ are independent variables and $F$ is the field of quotients of the polynomial rings $\mathbb F_2[x,y]$. Find a finite extension $L$ of $F$ such that it has infinitely many intermediate subfields containing $F$.

I have tried a lot but cannot give an example. Thank you a lot.

Answer 1

Take $L=\mathbb F_2(\sqrt x,\sqrt y)$.