A counterexample to the Fourteenth Problem of Hilbert in dimension four

This paper presents a family of new counterexamples to Hilbertʹs Fourteenth Problem. They are realized as subfields of the rational function field of four variables over a field of characteristic zero of transcendence degree three. It was previously not known whether any counterexample could be found as a subfield of a rational function field of four variables, or whether counterexamples with minimal transcendence degree three could exist.