Hilbert's basis theorem

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In mathematics Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology).

In modern algebra, rings whose ideals have this property are called Noetherian rings. Every field, and the ring of integers are Noetherian rings. So, the theorem can be generalized and restated as: every polynomial ring over a Noetherian ring is also Noetherian.

The theorem was stated and proved by David Hilbert in 1890 in his seminal article on invariant theory[1], where he solved several problems on invariants. In this article, he proved also two other fundamental theorems on polynomials, the Nullstellensatz (zero-locus theorem) and the syzygy theorem (theorem on relations). These three theorems were the starting point of the interpretation of algebraic geometry in terms of commutative algebra. In particular, the basis theorem implies that every algebraic set is the intersection of a finite number of hypersurfaces.

Another aspect of this article had a great impact on mathematics of the 20th century; this is the systematic use of non-constructive methods. For example, the basis theorem asserts that every ideal has a finite generator set, but the original proof does not provide any way to compute it for a specific ideal. This approach was so astonishing for mathematicians of that time that the first version of the article was rejected by Paul Gordan, the greatest specialist of invariants of that time, with the comment "This is not mathematics. This is theology."[2] Later, he recognized "I have convinced myself that even theology has its merits."[3]

Statement

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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.[4]

Corollary. If   is a Noetherian ring, then   is a Noetherian ring.

Hilbert proved the theorem (for the special case of multivariate polynomials over a field) in the course of his proof of finite generation of rings of invariants.[1] The theorem is interpreted in algebraic geometry as follows: every algebraic set is the set of the common zeros of finitely many polynomials.

Hilbert's proof is highly non-constructive: it proceeds by induction on the number of variables, and, at each induction step use the non-constructive proof for one variable less. Introduced more than eighty years later, Gröbner bases allow a direct proof that is as constructive as possible: Gröbner bases produce an algorithm for testing whether a polynomial belong to the ideal generated by other polynomials. So, given an infinite sequence of polynomials, one can construct algorithmically the list of those polynomials that do not belong to the ideal generated by the preceding ones. Gröbner basis theory implies that this list is necessarily finite, and is thus a finite basis of the ideal. However, for deciding whether the list is complete, one must consider every element of the infinite sequence, which cannot be done in the finite time allowed to an algorithm.

Proof

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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

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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. By construction,   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

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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

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

  1. By induction we see that   will also be Noetherian.
  2. 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.
  3. 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

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

References

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  1. ^ a b Hilbert, David (1890). "Über die Theorie der algebraischen Formen". Mathematische Annalen. 36 (4): 473–534. doi:10.1007/BF01208503. ISSN 0025-5831. S2CID 179177713.
  2. ^ Reid 1996, p. 34.
  3. ^ Reid 1996, p. 37.
  4. ^ Roman 2008, p. 136 §5 Theorem 5.9

Further reading

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