In mathematics, Hilbert's basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. This can be translated into algebraic geometry as follows: every variety over k can be described as the set of common roots of finitely many polynomial equations.
Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.
A slightly more general statement of Hilbert's basis theorem is: if R is a left (respectively right) Noetherian ring, then the polynomial ring R[X] is also left (respectively right) Noetherian.
The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the HILBASIS file.
Reference
Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.