By this thread, if I have a subfield $k\subseteq F\subseteq k(x_1,\dots,x_n)$, $F$ is of the form $F=k(\phi_1,\dots,\phi_m)$ for some rational functions $\phi_1,\dots,\phi_m\in k(x_1,\dots,x_n)$. But it is not always the case that $F=k(f_1,\dots,f_m)$ for polynomials $f_1,\dots,f_m\in k[x_1,\dots,x_n]$, right? I thought of taking e.g. $F=\mathbb{Q}(\frac{x}{x^2+1})\subseteq\mathbb{Q}(x)$, since I don't see any way of writing $F=\mathbb{Q}(p)$ for a polynomial $p\in\mathbb{Q}[x]$. Is there one, and if yes, is it always possible to do so?
Since I think it doesn't work that way, and since I'm originally interested in the intersection $F\cap k[x_1,\dots,x_n]$ : Can I somehow classify the rational functions $\phi$ such that $k(\phi)$ can not be written as $k(f_1,\dots,f_m)$ for polynomials $f_i$? Or such that the intersection $k(\phi)\cap k[x_1,\dots,x_n]=k$? For instance, rational functions that can be written as $\frac{1}{p}$ for a polynomial $p$ are none of these, since $k(\frac{1}{p})=k(p)$. For simplicity, I'm thinking of the case of a 'nice' field $k$ here, like $k=\mathbb{Q}$ or $\mathbb{C}$. Also, (reduced) rational functions $\frac{f}{g}$ with $\deg(f),\deg(g)\neq 0$ and $\deg(f)\neq\deg(g)$ should be examples, while the case of same degree seems to include examples for both sides...
If $F=k(\phi_1,\dots,\phi_m)$, and if I knew that $\phi_1,\dots,\phi_r$ were in the above 'class', then $F\cap k[x_1,\dots,x_n]$ should be equal to $k(\phi_{r+1},\dots,\phi_m)\cap k[x_1,\dots,x_n]$. Then I could write this intersection as $k(f_1,\dots,f_m)\cap k[x_1,\dots,x_n]$ for polynomials $f_i$.
Question edited in: To make it more precise, because I figured it could be unclear from my vague descriptions above: The question that interests me the most is the following:
If $F$ is any such subfield, can I always write the intersection $F\cap k[x_1,\dots,x_n]$ as $k(f_1,\dots,f_r)\cap k[x_1,\dots,x_n]$ for polynomials $f_1,\dots,f_r\in k[x_1,\dots,x_n]$?
Somehow I feel I'm getting lost in a matter which should be simpler. Any suggestions, references or answers are very welcome, especially regarding the intersection with the polynomial ring, since the thoughts before that arose from this problem.