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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
Hilbert's basis theorem asserts that every ideal of a
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
over a field has a finite
generating set In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to ...
(a finite ''basis'' in Hilbert's terminology). In modern
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, rings whose ideals have this property are called
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
s. Every field, and the ring of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s 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 David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
in 1890 in his seminal article on
invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...
, 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 Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
in terms of
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
. In particular, the basis theorem implies that every algebraic set is the intersection of a finite number of
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
s. 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 Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous ...
, the greatest specialist of invariants of that time, with the comment "This is not mathematics. This is theology." Later, he recognized "I have convinced myself that even theology has its merits."


Statement

If R is a ring, let R /math> denote the ring of
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s in the indeterminate X over R.
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...
proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R /math>. Formally,
Hilbert's Basis Theorem. If R is a Noetherian ring, then R /math> is a Noetherian ring.
Corollary. If R is a Noetherian ring, then R _1,\dotsc,X_n/math> is a Noetherian ring.
Hilbert proved the theorem (for the special case of
multivariate polynomial In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative intege ...
s over a field) in the course of his proof of finite generation of rings of invariants. The theorem is interpreted in
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
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 uses 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

Theorem. If R is a left (resp. right)
Noetherian ring In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
, then the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
R /math> 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 \mathfrak a \subseteq R /math> is a non-finitely generated left ideal. Then by recursion (using the
axiom of dependent choice In mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
) there is a sequence of polynomials \ such that if \mathfrak b_n is the left ideal generated by f_0, \ldots, f_ then f_n \in \mathfrak a \setminus \mathfrak b_n is of minimal degree. By construction, \ is a non-decreasing sequence of
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s. Let a_n be the leading coefficient of f_n and let \mathfrak be the left ideal in R generated by a_0,a_1,\ldots. Since R is Noetherian the chain of ideals :(a_0)\subset(a_0,a_1)\subset(a_0,a_1,a_2) \subset \cdots must terminate. Thus \mathfrak b = (a_0,\ldots ,a_) for some
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
N. So in particular, :a_N=\sum_ u_a_, \qquad u_i \in R. Now consider :g = \sum_u_X^f_, whose leading term is equal to that of f_N; moreover, g\in\mathfrak b_N. However, f_N \notin \mathfrak b_N, which means that f_N - g \in \mathfrak a \setminus \mathfrak b_N has degree less than f_N, contradicting the minimality.


Second proof

Let \mathfrak a \subseteq R /math> be a left ideal. Let \mathfrak b be the set of leading coefficients of members of \mathfrak a. This is obviously a left ideal over R, and so is finitely generated by the leading coefficients of finitely many members of \mathfrak a; say f_0, \ldots, f_. Let d be the maximum of the set \, and let \mathfrak b_k be the set of leading coefficients of members of \mathfrak a, whose degree is \le k. As before, the \mathfrak b_k are left ideals over R, and so are finitely generated by the leading coefficients of finitely many members of \mathfrak a, say :f^_, \ldots, f^_ with degrees \le k. Now let \mathfrak a^*\subseteq R /math> be the left ideal generated by: :\left\\!\!\;. We have \mathfrak a^*\subseteq\mathfrak a and claim also \mathfrak a\subseteq\mathfrak a^*. Suppose for the sake of contradiction this is not so. Then let h\in \mathfrak a \setminus \mathfrak a^* be of minimal degree, and denote its leading coefficient by a. :Case 1: \deg(h)\ge d. Regardless of this condition, we have a\in \mathfrak b, so a is a left linear combination ::a=\sum_j u_j a_j :of the coefficients of the f_j. Consider ::h_0 =\sum_u_X^f_, :which has the same leading term as h; moreover h_0 \in \mathfrak a^* while h\notin\mathfrak a^*. Therefore h - h_0 \in \mathfrak a\setminus\mathfrak a^* and \deg(h - h_0) < \deg(h), which contradicts minimality. :Case 2: \deg(h) = k < d. Then a\in\mathfrak b_k so a is a left linear combination ::a=\sum_j u_j a^_j :of the leading coefficients of the f^_j. Considering ::h_0=\sum_j u_j X^f^_, :we yield a similar contradiction as in Case 1. Thus our claim holds, and \mathfrak a = \mathfrak a^* which is finitely generated. Note that the only reason we had to split into two cases was to ensure that the powers of X multiplying the factors were non-negative in the constructions.


Applications

Let R be a Noetherian
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
. Hilbert's basis theorem has some immediate
corollaries In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
. #By induction we see that R _0,\dotsc,X_/math> will also be Noetherian. #Since any
affine variety In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...
over R^n (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal \mathfrak a\subset R _0, \dotsc, X_/math> 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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
of finitely many
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
s. #If A is a finitely-generated R-algebra, then we know that A \simeq R _0, \dotsc, X_/ \mathfrak a, where \mathfrak a is an ideal. The basis theorem implies that \mathfrak a must be finitely generated, say \mathfrak a = (p_0,\dotsc, p_), i.e. A is finitely presented.


Formal proofs

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


References


Further reading

* Cox, Little, and O'Shea, ''Ideals, Varieties, and Algorithms'', Springer-Verlag, 1997. * The definitive English-language biography of Hilbert. *{{citation , last=Roman , first=Stephen , title=Advanced Linear Algebra , edition=Third , series=
Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...
, publisher = Springer , date=2008, pages= , isbn=978-0-387-72828-5 , author-link=Steven Roman Commutative algebra Invariant theory Articles containing proofs Theorems in ring theory David Hilbert Theorems about polynomials