Existing in Mathlib · Algebra
Hilbert Basis Theorem
If every ideal in a commutative ring can be generated by finitely many elements, the same remains true after adjoining one polynomial variable.
- Hilbert basis theorem
- Noetherian rings
- univariate polynomial rings
- finitely generated ideals
- leading coefficients
- degree induction
Exact theorem
Exact Mathlib statement
protected theorem Polynomial.isNoetherianRing {R : Type u} [CommRing R] [inst : IsNoetherianRing R] : IsNoetherianRing R[X]The theorem at a glance
Hilbert Basis Theorem — finite generation survives one polynomial variable
A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.
Hilbert Basis Theorem — finite generation survives one polynomial variable
Detailed visual description
The poster centers the exact transfer R Noetherian to R[X] Noetherian. Its engraved proof route tracks leading coefficients by degree, marks a stabilization threshold, finitely generates the bounded-degree part, and cancels higher leading terms to induct downward in degree.
Statement structure
Statement and scope
Read the exact Mathlib declaration
This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.
Hilbert Basis Theorem — scientific diagram
![A lattice marked R feeds a five-stage ascending stack whose upper two stages lock into a gold stabilization bracket; three generator nodes gather polynomial strands, and four progressively shortened curves descend into a finite enclosure before the endpoint R[X].](/assets/theorem-visual-styles/scientific-v1/mathlib/hilbert-basis-theorem/theorem-schematic-scientific-v2.png)
Detailed visual description
The diagram follows one coefficient-ring network into an ascending family of leading-coefficient stages, with the upper stages sharing a gold stabilization enclosure. Three schematic generator nodes gather strands into one vertical polynomial-degree direction. Four successively shortened curves show leading-term cancellation, and three remaining strands lie in a finite-span enclosure before the single endpoint R[X]. The displayed counts are illustrative, not generator or degree bounds.
Why it matters
A mathematical landmark
Hilbert's basis theorem is a foundational finiteness result in commutative algebra: adjoining one polynomial variable preserves the Noetherian condition and keeps every polynomial ideal finitely generated.
ProofAtlas record
What has been checked
Mathlib is the source of the theorem; the local Lean replay and page review are separate.
Claim boundary
No new theorem is claimed
This target indexes Mathlib's protected theorem that a commutative Noetherian ring R has a Noetherian univariate polynomial ring R[X]. The conclusion is the IsNoetherianRing R[X] typeclass proposition, and the source registers the theorem as an instance on the following line. It is not a multivariate endpoint, a generator-computation algorithm, or new ProofAtlas mathematics.
- The exact declaration adjoins one polynomial variable; it does not itself state a multivariable polynomial theorem.
- The theorem produces a typeclass proposition, not an explicit algorithm for computing generators of an ideal.
- The source assumes a commutative ring; this page does not extend the result to arbitrary semirings or noncommutative polynomial constructions.
- ProofAtlas is indexing an existing Mathlib theorem, not claiming mathematical novelty.
Source and local evidence
Where the theorem comes from
- Existing declaration
Polynomial.isNoetherianRingin mathlib- Relationship
- The checked artifact is the upstream declaration itself
- Local Lean evidence
artifact.library.mathlib.hilbert-basis-theorem.v001