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Hilbert's Basis TheoremIn 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 kx2, ..., 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 Grbner 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 RX 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.
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