Zariski's finiteness theorem

In algebra, Zariski's finiteness theorem gives a positive answer to Hilbert's 14th problem for the polynomial ring in two variables, as a special case.{{cite web|url=http://aix1.uottawa.ca/~ddaigle/articles/H14survey.pdf|title=HILBERT’S FOURTEENTH PROBLEM AND LOCALLY NILPOTENT DERIVATIONS|access-date=2023-08-25}} Precisely, it states:

:Given a normal domain A, finitely generated as an algebra over a field k, if L is a subfield of the field of fractions of A containing k such that the transcendence degree $\backslash operatorname\{tr.deg\}\_k(L)\; \backslash le\; 2$, then the k-subalgebra $L\; \backslash cap\; A$ is finitely generated.