Hilbert's fourteenth problem

Remarks

The motivating case is a ring of invariants: if a group $G$ acts linearly on $k^n$, then $k[x_1, \ldots, x_n]^G = k(x_1, \ldots, x_n)^G \cap k[x_1, \ldots, x_n]$. The answer is no in general. Nagata gave a counterexample in 1959, as the ring of invariants of a linear action of a group on a polynomial ring. The answer is yes in several important cases. If $G$ is a reductive group, the ring of invariants of a rational action on $k[x_1, \ldots, x_n]$ is finitely generated. In characteristic $0$ this goes back to Hilbert's finiteness theorem of 1890. Zariski proved that the answer is yes when $K$ has transcendence degree at most $2$ over $k$. Weitzenböck proved that, in characteristic $0$, the invariants of a linear action of the additive group of $k$ are finitely generated.