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, ..., x_n] is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over k can be described as the set of common roots of finitely many polynomial equations. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... David Hilbert (January 23, 1862, Königsberg, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... For the toll-free telephone number see Toll-free telephone number Year 1888 (MDCCCLXXXVIII) was a leap year starting on Sunday (click on link for calendar) of the Gregorian calendar (or a leap year starting on Friday of the 12-day slower Julian calendar). ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a module is a finitely-generated module if it has a finite generating set. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K, of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 and so on up to Pm(X1, ...,Xn) = 0 for some integer m. ...

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. Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a finite list of well-defined instructions for accomplishing some task that, given an initial state, will terminate in a defined end-state. ... In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of: the Euclidean algorithm for computation of univariate greatest common...

Proof

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. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

Proof: For , if with a_n not equal to 0, then degf: = n and a_n is the leading coefficient of f. Let I be an ideal in R[x] and assume I is not finitely generated. Then inductively construct a sequence f₁,f₂,... of elements of I such that f_{i + 1} has minimal degree among elements of , where J_i is the ideal generated by f₁,...,f_i. Let a_i be the leading coeffecient of f_i and let J be the ideal of R generated by the sequence a₁,a₂,.... Since R is Noetherian there exists N such that J is generated by a₁,...,a_N. Therefore for some . We obtain a contradiction by considering where n_i = degf_{N + 1} − degf_i, because degg = degf_{N + 1} and their leading coefficients agree, so that f_{N + 1} − g has degree strictly less than degf_{N + 1}, contradicting the choice of f_{N + 1}. Thus I is finitely generated. Since I was an arbitrary ideal in R[x], every ideal in R[x] is finitely generated and R[x] is therefore Noetherian.

or:

Given an ideal J of R[X] let L be the set of leading coefficients of the elements of J. Then L is clearly an ideal in R so is finitely generated by a(1),...,a(n) in L, and there are f(1),...,f(n) in J with a(i) being the leading coefficient of f(i). Let d(i) be the degree of f(i) and let N be the maximum of the d(i)'s. Now for each k=0,...,N-1 let L(k) be the set of leading coeficients of elements of J with 0. Then again, L(k) is clearly an ideal in R so is finitely generated by a(k,1),...,a(k,m(k)) say. As before, let f(k,i) in J have leading coefficient a(k,i). Let H be the ideal in R[X] generated by the f(i)'s and f(k,i)'s. Then surely H is contained in J and assume there is an element f in L not belonging to H, of least degree d, and leading coefficient a. If d is larger or equal to N then a is in L so, a=r(1)a(1)+...+r(1)a(n) and g= r(1)X^{d − d(1)} + ... + r(n)X^{d − d(n)} is a polynomial of same degree as f with the same leading coefficient, hence f-g is in J and by minimality of f, f-g is in H so f is in H, a contradiction. If, on the other hand, d is strictly smaller than N, then a is in L(d) so a=r(1)a(d,1)+...+r(m(d))a(d,m(d)) and g=r(1)f(d,1)+...+r(m(d))f(d,m(d)) is a polynomial in H of the same degree as f and with the same leading coefficient. Hence, f-g is in H by minimality of f, and so f is in H, a contradiction. Then H=J and so J is finitely generated, and consequently R[X] is Noetherian.

Other

The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the HILBASIS file. The 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 to and used in new articles. ...

References

• Cox, Little, and O'Shea, Ideals, Varieties, and Algorithms, Springer-Verlag, 1997.