Krull showed that every 1-dimensional normal Noetherian local ring is analytically unramified; more precisely he showed that a 1-dimensional normal Noetherian local domain is analytically unramified if and only if its integral closure is a finite module.
[citation needed] This prompted Zariski (1948) to ask whether a local Noetherian domain such that its integral closure is a finite module is always analytically unramified.
The second says that a Noetherian local domain is analytically unramified if and only if, for every finitely-generated R-algebra S lying between R and the field of fractions K of R, the integral closure of S in K is a finitely generated module over S. The second follows from the first.
Let T be the subring of the formal power series ring K [[x,y]] generated by K and K2 [[x,y]] and the element Σ(unxn+ vnyn).
Nagata proves that T is a normal local noetherian domain whose completion has nonzero nilpotent elements, so T is analytically ramified.