If $F$ is any algebraically closed field, and $L \supset F(X)$ is a finite extension of the purely transcendental extension of $F$ of transcendence degree $1$, then can $L$ necessarily be embedded into $F(Y)$ (so that $X$ becomes a rational function in $Y$)?
It seems like this ought to be an easy question to google, but I haven't found anything. I did find this wikipedia article which indicates a connection to algebraic geometry, so if my grasp of algebraic geometry were strong enough, I would probably be able to answer this question on my own. Alas, I am not there yet. :)