# Lüroth's theorem for transcendence degree two

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!

• A term to search: "Castelnuovo's theorem". – KReiser May 6 '18 at 2:05
• Thank you very much; I will search "Castelnuovo's theorem" (it seems to appear in the notes I mentioned above, on page 8). Please, do you know the answer to my second question (2)? – user237522 May 6 '18 at 2:13