Hilbert Basis TheoremEdit
The Hilbert Basis Theorem is a foundational result in modern algebra, tying finiteness notions to the structure of polynomial rings. Named after the German mathematician David Hilbert, it sits at the crossroads of Noetherian ring theory, ring theory and algebraic geometry. The theorem asserts that certain rings retain a crucial finiteness property when one forms polynomial extensions, ensuring that the algebraic description of geometric objects defined by polynomial equations remains manageable.
In its most quoted form, the theorem says that if a ring R is Noetherian, then the polynomial ring R[x1, ..., xn] is also Noetherian. Equivalently, every ideal in R[x1, ..., xn] is finitely generated. This finiteness phenomenon is the engine behind many constructive methods in mathematics: one can always obtain a finite set of generators for the objects defined by polynomial equations, which in turn underpins algorithmic approaches in computational algebra and the study of algebraic varieties.
Statement
- Noetherian preservation under polynomial extension: If R is a Noetherian ring, then the polynomial ring R[x1, ..., xn] is Noetherian for any finite list of indeterminates. This is typically proved by induction on the number of variables.
- Finite generation of ideals: If R is Noetherian and I is an ideal of R[x1, ..., xn], then I can be generated by finitely many elements of R[x1, ..., xn].
From these core points follow several practical corollaries. For instance, any finitely generated R-algebra A = R[x1, ..., xn]/I is Noetherian, and whenever R is Noetherian, polynomial rings over R have a well-behaved ideal structure. In the special case where R is a field k, the theorem implies that k[x1, ..., xn] is Noetherian, and hence any finitely generated k-algebra is Noetherian.
Historical context also clarifies the motivation. Hilbert introduced and used the theorem in the context of invariant theory, showing that the ring of invariants under a finite group action is finitely generated. This resolved long-standing questions about finiteness and helped establish a robust framework for studying solutions to systems of polynomial equations.
Proof ideas
Several routes to the Hilbert Basis Theorem exist, reflecting the theorem’s central role in algebra. A common approach uses induction on the number of indeterminates:
- Base case: R Noetherian implies R[x] is Noetherian. The key is to manage the degrees of polynomials and to show that any ascending chain of ideals stabilizes.
- Inductive step: Assume R[x1, ..., x_{n-1}] is Noetherian. For an ideal I ⊆ R[x1, ..., xn], consider, for each degree d, the set of coefficients of x_n^d that appear in polynomials of I. These coefficients generate an ascending chain of ideals in the subring R[x1, ..., x_{n-1}], which stabilizes by the inductive hypothesis. From there, one extracts a finite set of generators for I, thus showing I is finitely generated.
Other proofs use Gröbner bases and leading-term orders, or rely on more algebraic descriptions such as Dickson’s lemma in combinatorial commutative algebra. The diversity of proofs underlines the theorem’s foundational nature and its connections to computational methods.
Consequences and applications
- Finite descriptions of algebraic objects: The Noetherian property guarantees that many algebraic constructions arising from polynomial equations are finitely generated, which is essential for both theoretical work and computations.
- Algebraic geometry: Hilbert’s Basis Theorem underpins the statement that affine algebraic sets over a field can be described by finitely many polynomials; this finiteness is central to the computational and structural study of varieties.
- Finitely generated algebras: If a ring R is Noetherian, then any finitely generated R-algebra is Noetherian, which broadens the scope of Noetherian techniques to a wide class of algebraic objects.
- Computational algebra: The theorem provides a theoretic justification for terminating procedures that compute generating sets of ideals or invariants, and it anchors algorithms in Gröbner basis theory and related methods.
Examples
- Over a field k, the ring k[x] is Noetherian, and thus k[x1, ..., xn] is Noetherian for any finite n. Consequently, any ideal in k[x1, ..., xn] is finitely generated.
- In a broader setting, if R is a Noetherian ring (for example, a finitely generated algebra over a field), then the polynomial ring R[x1, ..., xn] inherits the Noetherian property, carrying finiteness from the base ring to polynomial extensions.