Hilberts basis theorem

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, ..., x_n] 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 polynomials.

Hilbert's innovative proof is a proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finite list of basis polynomials: it only shows that they must exist.