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, 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." Later, he recognized "I have convinced myself that even theology has its merits."
Statement If R {\displaystyle R} is a ring, let R [ X ] {\displaystyle R[X]} denote the ring of polynomials in the indeterminate X {\displaystyle X} over R {\displaystyle R} . Hilbert proved that if R {\displaystyle R} is "not too large", in the sense that if R {\displaystyle R} is Noetherian, the same must be true for R [ X ] {\displaystyle R[X]} . Formally,
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. 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 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 belongs 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
We will give two proofs, in both only the "left" case is considered; the proof for the right case is similar.
First proof Suppose a ⊆ R [ X ] {\displaystyle {\mathfrak {a}}\subseteq R[X]} is a non-finitely generated left ideal. Then by recursion (using the axiom of dependent choice) there is a sequence of polynomials { f 0 , f 1 , … } {\displaystyle \{f_{0},f_{1},\ldots \}} such that if b n {\displaystyle {\mathfrak {b}}_{n}} is the left ideal generated by f 0 , … , f n − 1 {\displaystyle f_{0},\ldots ,f_{n-1}} then f n ∈ a ∖ b n {\displaystyle f_{n}\in {\mathfrak {a}}\setminus {\mathfrak {b}}_{n}} is of minimal degree. By construction, { deg ( f 0 ) , deg ( f 1 ) , … } {\displaystyle \{\deg(f_{0}),\deg(f_{1}),\ldots \}} is a non-decreasing sequence of natural numbers. Let a n {\displaystyle a_{n}} be the leading coefficient of f n {\displaystyle f_{n}} and let b {\displaystyle {\mathfrak {b}}} be the left ideal in R {\displaystyle R} generated by a 0 , a 1 , … {\displaystyle a_{0},a_{1},\ldots } . Since R {\displaystyle R} is Noetherian the chain of ideals
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