Given algebraically closed field $\mathbb{F}$ and rational function in 2 variables $f(x,y)\in \mathbb{F}(x,y)$, does it factor out in linear terms $(ax+by), a,b\in \mathbb{F}$, or in other terms, is subalgebra generated by $S=\{(ax+by)^{\pm1}; a,b\in\mathbb{F}\}$ the whole space?
$$(S)= \mathbb{F}(x,y)$$
Does this at least hold in $\mathbb{C}$? Is there some general theorem which proves this for arbitrary amount of variables? If so, please refer me to the book as I would like to see the proof