Zariski's finiteness theorem explained

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

\operatorname{tr.deg} _{k(L)} \le 2 </Math>, then the ''k'' - subalgebra <math>L \cap A is finitely generated.

References

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

Notes and References

1. Web site: HILBERT’S FOURTEENTH PROBLEM AND LOCALLY NILPOTENT DERIVATIONS. 2023-08-25.