The List of Hilbert’s Twenty-Three Problems: Hilbert’s Problem #14

Finiteness of Certain Systems of Functions: Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated

The motivation for Hilbert’s 14th problem came from previous work he had done showing that algebraic structures called rings arising in a particular way from larger structures must be finitely generated; that is, they could be described using only a finite number of building blocks. Hilbert asked whether the same was true for a broader class of rings. In 1958 Masayoshi Nagata resolved the question by finding a counterexample.