of $\Bbbk (x_{1},\dots,x_{n})$: $\Bbbk  \subset K \subset
\Bbbk (x_{1},\dots,x_{n})$. Is the ring $K \cap \Bbbk [x_{1},\dots,x_{n}]$ finitely generated over $\Bbbk $?”.

This problem of the [*finite generation of special subalgebras of the polynomial algebra $\Bbbk [x_{1},\dots,x_{n}]$*]{} is known as [*Hilbert’s 14$^{\text{th}}$ problem*]{} because it appeared with that number in the list of 23 problems presented by Hilbert in the International Congress of Mathematicians celebrated in Paris in 1900 ([@kn:hilbertpr]).

A particularly important case is the following:

Lt $G\subset  \operatorname{GL}_{n}$ be a subgroup, consider the induced action of $G$ on $\Bbbk [x_{1},\dots,x_{n}]$ and call $K={}{^G}{\Bbbk (x_{1},\dots,x_{n})}{}$. As ${}{^G}{\Bbbk [x_{1},\dots,x_{n}]}{} = K 
\cap \Bbbk [x_{1},\dots,x_{n}]$, the finite generation of rings of invariants could –in principle– be deduced from an affirmative answer to Hilbert’s problem.

In 1900, when Hilbert formulated his 14$^{\text{th}}$ problem, a few particular cases were already solved. Classical invariant theorists were concerned with the invariants of “quantics” (invariants for certain actions of ${\operatorname{SL}}_{m}(\mathbb C)$). In this situation the finite generation was proved by Gordan in 1868 for $m=2$ and by Hilbert in 1890 for arbitrary $m$. Hilbert mentioned as motivation for his 14$^{\text{th}}$ problem work by Hurwitz and also by Maurer–that turned out to be partially incorrect–.

Maurer’s work contains 