Hilbert's 14th problem asks whether certain subrings are finitely generated. The subring is generated as follows: Let k be a field. K is defined as subfield of the rational functions in n variables (x1, ..., xn) over k. Let the ring be K (intersect) k[x1, ..., xn]. Hilbert conjectured that all such subrings are finitely generated. In 1959, Masayoshi Nagata found a counterexample to Hilbert's conjecture.

1. M. Nagata: Lectures on the fourteenth problem of Hilbert. Lect. Notes 31, Tata Inst. Bombay, 1965