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.

References

1. "HILBERT'S FOURTEENTH PROBLEM AND LOCALLY NILPOTENT DERIVATIONS" (PDF). Retrieved 2023-08-25.

• Zariski, O. (1954). "Interprétations algébrico-géométriques du quatorzième problème de Hilbert". Bull. Sci. Math. (2). 78: 155–168.