In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.

Statement
If  is a ring, let  denote the ring of polynomials in the indeterminate  over . Hilbert proved that if  is "not too large", in the sense that if  is Noetherian, the same must be true for . Formally,

Hilbert's Basis Theorem. If  is a Noetherian ring, then  is a Noetherian ring.
Corollary. If  is a Noetherian ring, then  is a Noetherian ring.

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. Hilbert proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.

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
Theorem. If  is a left (resp. right) Noetherian ring, then the polynomial ring  is also a left (resp. right) Noetherian ring.

Remark. We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.

First proof
Suppose  is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials  such that if  is the left ideal generated by  then  is of minimal degree. It is clear that  is a non-decreasing sequence of natural numbers. Let  be the leading coefficient of  and let  be the left ideal in  generated by . Since  is Noetherian the chain of ideals

must terminate. Thus  for some integer . So in particular,

Now consider

whose leading term is equal to that of ; moreover, . However, , which means that  has degree less than , contradicting the minimality.

Second proof 
Let  be a left ideal. Let  be the set of leading coefficients of members of . This is obviously a left ideal over , and so is finitely generated by the leading coefficients of finitely many members of ; say . Let  be the maximum of the set , and let  be the set of leading coefficients of members of , whose degree is . As before, the  are left ideals over , and so are finitely generated by the leading coefficients of finitely many members of , say

with degrees . Now let  be the left ideal generated by:

We have  and claim also . Suppose for the sake of contradiction this is not so. Then let  be of minimal degree, and denote its leading coefficient by .

Case 1: . Regardless of this condition, we have , so  is a left linear combination

of the coefficients of the . Consider

which has the same leading term as ; moreover  while . Therefore  and , which contradicts minimality.

Case 2: . Then  so  is a left linear combination

of the leading coefficients of the . Considering

we yield a similar contradiction as in Case 1.

Thus our claim holds, and  which is finitely generated.

Note that the only reason we had to split into two cases was to ensure that the powers of  multiplying the factors were non-negative in the constructions.

Applications 
Let  be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries.

By induction we see that  will also be Noetherian. 
Since any affine variety over  (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal  and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces. 
If  is a finitely-generated -algebra, then we know that , where  is an ideal. The basis theorem implies that  must be finitely generated, say , i.e.  is finitely presented.

Formal proofs
Formal proofs of Hilbert's basis theorem have been verified through the Mizar project (see HILBASIS file) and  Lean (see ring_theory.polynomial).

References

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

Commutative algebra
Invariant theory
Articles containing proofs
Theorems in ring theory
David Hilbert