# Hilbert's basis theorem

﻿
Hilbert's basis theorem

In mathematics, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a field is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. The theorem is named for the German mathematician David Hilbert who first proved it in 1888.

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.

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.

Proof: For $f in R \left[x\right]$, if $f=sum_\left\{k=0\right\}^na_kx^k$ with $a_n$ not equal to "0", then $deg f := n$ 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$f_1,f_2,...$ of elements of "I" such that $f_\left\{i+1\right\}$ has minimal degree among elements of $Isetminus J_i$, where $J_i$ is the ideal generated by $f_1,...,f_i$. Let $a_i$ be the leading coeffecient of $f_i$ and let "J" be the ideal of "R" generated by the sequence $a_1,a_2,...$. Since "R" is Noetherian there exists "N" such that "J" is generated by $a_1,...,a_N$. Therefore $a_\left\{N+1\right\} = sum_\left\{i=1\right\}^Nu_ia_i$ for some $u_1,...,u_N in R$. We obtain a contradiction by considering $g = sum_\left\{i=1\right\}^Nu_if_ix^\left\{n_i\right\}$ where $n_i = deg f_\left\{N+1\right\} - deg f_i$, because $deg g = deg f_\left\{N+1\right\}$ and their leading coefficients agree, so that $f_\left\{N+1\right\} - g$ has degree strictly less than $deg f_\left\{N+1\right\}$, contradicting the choice of $f_\left\{N+1\right\}$. 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 a constructive proof:

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 degree atmost k. 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 J 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= $r\left(1\right)X^\left\{d-d\left(1\right)\right\}f\left(1\right)+...+r\left(n\right)X^\left\{d-d\left(n\right)\right\}f\left(n\right)$ is of the same degree as f and has the same leading coefficient. Since g is in H, f-g is not,which contradicts the minimality of f. 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)). A similar construction as above again gives the same contradiction. Thus, J=H, which is finitely generated. QED.

Other

The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the [http://www.mizar.org/JFM/Vol12/hilbasis.html HILBASIS file] .

References

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Basis theorem — can refer to:* Hilbert s basis theorem, in algebraic geometry * Low basis theorem, in computability theory …   Wikipedia

• Hilbert's Nullstellensatz — (German: theorem of zeros, or more literally, zero locus theorem – see Satz) is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of… …   Wikipedia

• Hilbert basis — may refer to * Orthonormal basis * Hilbert basis (linear programming) * Hilbert s basis theorem …   Wikipedia

• Hilbert, David — born Jan. 23, 1862, Königsberg, Prussia died Feb. 14, 1943, Göttingen, Ger. German mathematician whose work aimed at establishing the formalistic foundations of mathematics. He finished his Ph.D. at the University of Königsberg (1884) and moved… …   Universalium

• Hilbert's fourteenth problem — In mathematics, Hilbert s fourteenth problem, that is, number 14 of Hilbert s problems proposed in 1900, asks whether certain rings are finitely generated. The setting is as follows: Assume that k is a field and let K be a subfield of the field… …   Wikipedia

• Basis (linear algebra) — Basis vector redirects here. For basis vector in the context of crystals, see crystal structure. For a more general concept in physics, see frame of reference. In linear algebra, a basis is a set of linearly independent vectors that, in a linear… …   Wikipedia

• Hilbert space — For the Hilbert space filling curve, see Hilbert curve. Hilbert spaces can be used to study the harmonics of vibrating strings. The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It… …   Wikipedia

• Hilbert–Speiser theorem — In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any abelian extension K of the rational field Q . The Kronecker–Weber theorem… …   Wikipedia

• Hilbert–Schmidt operator — In mathematics, a Hilbert–Schmidt operator is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm, meaning that there exists an orthonormal basis {e i : i in I} of H with the property:sum {iin I} |Ae i|^2 < infty. If this… …   Wikipedia

• Theorem — The Pythagorean theorem has at least 370 known proofs[1] In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements …   Wikipedia