Classical affine algebraic varieties appear as sets of zeros of a set of polynomials in affine -dimensional space over a field . The coordinate algebra of is the algebra of polynomials in variables, , and the coordinate algebra of an affine algebraic variety is where is the ideal generated by .
The Hilbert basis theorem (HBT) asserts that this ideal is finitely generated; and consequently is a noetherian ring. For a proof see standard textbooks on commutative algebra or algebraic geometry (e.g. Atiyah, MacDonald); there is also a proof on wikipedia.
More generally, a finitely generated commutative algebra over a commutative noetherian ring is noetherian. For the case of a field, this is the case in the previous paragraphs.
More at Noetherian ring.
The theorem vastly generalises and subsumes Paul Gordon?’s work on invariant theory, albeit in a non-constructive way. Emmy Noether wrote a short paper in 1920 that sidestepped the use of the HBT to construct a basis for, and so implying the finite generation of, a certain ring of invariants attached to any finite group.