Home > Hilbert's basis theorem
Commutative algebra Ring theory TheoremsIn 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 basesIn computer algebra and computational algebraic geometry, a Grobner basis (named after Wolfgang Grobner) is a particular kind of generating subset of an ideal I in a polynomial ring. It is characterised by the property that the ideal given by the leading.
A slightly more general statement of Hilbert's basis theorem is: if R is a left (respectively right) Noetherian ringRing theory In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. Introduction Rings of polynomials over fields have many special properties; properties that follow from the fact that polynomial rings are, then the polynomial ringIn abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a commutative ring. More precisely, let R be a commutative ring. The polynomial ring in n variables X . X is the set of all polynomials in those R[X] is also left (respectively right) Noetherian.
The Mizar projectThe Mizar system consists of a language for writing strictly formalized mathematical definitions and proofs, a computer program which is able to check proofs written in this language, and a library of definitions and proved theorems which can be referred 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.
