Let $R \subseteq \mathbb{C}(x,y)$ and assume that $R=\mathbb{C}(u,v)$, where $u,v \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$.
Here $\mathbb{N}$ includes $0$.
Assume that $R$ satisfies the following 'rare' property: For every monomial $x^iy^j$, $i \in \mathbb{N}$, $j \in \mathbb{N}$ (except the case $i=j=0$, for which we assume nothing), we have $\mathbb{C}(u,v,x^iy^j)=\mathbb{C}(x,y)$.
Question: Is it true that $R=\mathbb{C}(x,y)$?
I am not able to find a counterexample, but perhaps there is such.
I do not mind to further assume that $x+y$ also satisfies $\mathbb{C}(u,v,x+y)=\mathbb{C}(x,y)$.
Any help is welcome! Thank you very much.
Edit: Please, what about the following version of the above question:
Same as the above question, except that now the property does not include the diagonal, namely, $\mathbb{C}(u,v,x^ny^n)=\mathbb{C}(x,y)$, $n \geq 1$, may not hold.
I am not sure which of the following three options is better to assume for every $n \geq 1$:
- There is no information about adding $x^ny^n$ to $\mathbb{C}(u,v)$.
- $\mathbb{C}(u,v,x^ny^n) \subsetneq \mathbb{C}(x,y)$.
- $x^ny^n \in \mathbb{C}(u,v)$.
Perhaps the third option, $x^ny^n \in \mathbb{C}(u,v)$, has the following solution, based on the answer to the original question:
For a given field $F$ and a given pair $(i,j) \in \mathbb{N} \times \mathbb{N}$, denote by $C_{F,i,j}$ the condition/property that $F(x^iy^j)=\mathbb{C}(x,y)$, in other words, the condition that $x^iy^j$ is a primitive element for the extension $F \subseteq \mathbb{C}(x,y)$.
Then the answer shows the following: If $C_{R,i,j}$ holds for every $(i,j) \in \mathbb{N} \times \mathbb{N} - (0,0)$, then $[\mathbb{C}(x,y) : R]=2$.
(Trivially, if $C_{R,0,0}$ holds, then $R=\mathbb{C}(x,y)$, so in this case $[\mathbb{C}(x,y) : R ]=1$).
Now take $\tilde{R}=\mathbb{C}(\tilde{u},\tilde{v})$, where $\tilde{u}, \tilde{v} \in \mathbb{C}[x,y]$ are algebraically independent over $\mathbb{C}$, and assume that $C_{\tilde{R},i,j}$ holds for every $(i,j) \in \mathbb{N} \times \mathbb{N} - D$, where $D= \{(n,n)\}_{n \in \mathbb{N}}$.
Moreover, assume that for every $n \in \mathbb{N}$, $x^ny^n \in \tilde{R}$.
Claim: $\tilde{R}=\mathbb{C}(x,y)$ (or $[\mathbb{C}(x,y):\tilde{R}]\leq 2$?).
Proof of claim: Same as the answer, except that the last two lines are changed, the map is not surjective, $\tilde{R} \subseteq \mathbb{C}(x,y) \subseteq L$. Therefore, by considerations of degree extensions, $\tilde{R}=\mathbb{C}(x,y)$. Or maybe we just can conclude that $[\mathbb{C}(x,y):\tilde{R}]\leq 2$.
Maybe I am missing something.
Should I ask this as a new question, or is it ok to ask it here?