In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.
The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables,
- k(x1, ..., xn ) over k.
Consider now the k-algebra R defined as the intersection
![{\displaystyle R:=K\cap k[x_{1},\dots ,x_{n}]\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15d082eb2576f3e3e6d72d22a7c2832d65b0e028)
Hilbert conjectured that all such algebras are finitely generated over k.
Some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.