ABSTRACT
The original formulation by D. Hilbert of his famous 14th problem reads as follows: "By a finite field of integrality I mean a system of functions from which a finite number of functions can be chosen, in which all other functions of the system are rationally and integrally expressible. The problem amounts to this: to show that all relatively integral functions of any given domain of rationality always constitute a finite field of integrality." In 1900, when Hilbert formulated his 14th problem, a few particular cases were already solved. Hilbert mentioned as motivation for his 14th problem a paper by A. Hurwitz and also work by L. Maurer — that turned out to be partially incorrect. There are various counterexamples to Hilbert's original problem, and many of them seem to be based in Nagata's ideas.
