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.^[1] 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 ${\displaystyle \operatorname {tr.deg} _{k}(L)\leq 2}$, then the k-subalgebra ${\displaystyle L\cap A}$ is finitely generated.