Let $k$ be an algebraically closed field of characteristic zero, and $k \subsetneq L \subseteq k(x_1,\ldots,x_n)$ a field of transcendence degree two over $k$. According to the comments of ulrich and abx in this question, there exist $h_1,h_2 \in k(x_1,\ldots,x_n)$ such that $L=k(h_1,h_2)$.
(1) Could one please give a reference to a proof of this result, preferrably a pure algebraic proof.
(The proof here is not pure algebraic; of course, I mean the proof for transcendence degree two. The proof of transcendence degree one, presented in page 4, is pure algebraic).
Moreover, inspired by this question:
(2) Is it possible to find $h_1,h_2 \in k[x_1,\ldots,x_n]$?
Thank you very much!