Hilberts basis theorem

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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 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.