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. 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 t r . d e g k ( L ) ≤ 2 {\displaystyle \operatorname {tr.deg} _{k}(L)\leq 2} , then the k-subalgebra L ∩ A {\displaystyle L\cap A} is finitely generated.
References
Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.