Zariski's finiteness theorem

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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 [math]\displaystyle{ \operatorname{tr.deg}_k(L) \le 2 }[/math], then the k-subalgebra [math]\displaystyle{ L \cap A }[/math] is finitely generated.


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