Noether normalization lemma
E29377
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
All labels observed (6)
| Label | Occurrences |
|---|---|
| Noether normalization lemma canonical | 5 |
| Noether normalization | 1 |
| Noether's normalization lemma | 1 |
| Noether’s normalization lemma | 1 |
| Zariski’s lemma | 1 |
| graded Noether normalization | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T228994 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Noether normalization lemma Context triple: [Emmy Noether, notableWork, Noether normalization lemma]
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
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B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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D.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
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E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Noether normalization lemma Target entity description: The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
A.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
B.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
-
C.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
-
D.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in algebraic geometry ⓘ result in commutative algebra ⓘ |
| appearsIn |
standard graduate textbooks on algebraic geometry
ⓘ
standard graduate textbooks on commutative algebra ⓘ |
| appliesTo | finitely generated k-algebras ⓘ |
| assumes |
base field k
ⓘ
commutative rings with identity ⓘ |
| concerns |
affine schemes
ⓘ
affine varieties ⓘ finitely generated algebras over a field ⓘ integral extensions of rings ⓘ polynomial rings ⓘ |
| field |
algebraic geometry
ⓘ
commutative algebra ⓘ |
| framework |
affine algebraic geometry
ⓘ
scheme theory ⓘ |
| hasConsequence |
coordinate ring of an affine variety is module-finite over a polynomial subring
ⓘ
every affine variety over a field admits a finite dominant morphism to affine space of dimension equal to its Krull dimension ⓘ existence of generic linear projections with finite fibers ⓘ |
| hasVariant |
equivariant Noether normalization
ⓘ
Noether normalization lemma self-linksurface differs ⓘ
surface form:
graded Noether normalization
version for Noetherian rings ⓘ |
| holdsOver | any field ⓘ |
| implies |
Krull dimension of a finitely generated k-algebra equals the transcendence degree of its field of fractions over k
ⓘ
any affine k-algebra is a finite module over a polynomial subring ⓘ any affine variety admits a finite surjective morphism to an affine space of the same dimension ⓘ existence of a finite injective k-algebra homomorphism from a polynomial ring to a finitely generated k-algebra ⓘ |
| namedAfter | Emmy Noether ⓘ |
| relatedTo |
Hilbert basis theorem
ⓘ
Hilbert’s Nullstellensatz ⓘ Krull dimension ⓘ algebraic independence ⓘ finite morphisms of schemes ⓘ integral dependence ⓘ |
| states |
every finitely generated k-algebra is integral over a polynomial k-subalgebra
ⓘ
if A is a finitely generated k-algebra then there exist algebraically independent elements y1,…,yd in A such that A is integral over k[y1,…,yd] ⓘ |
| typicalProofUses |
induction on the number of generators
ⓘ
integral dependence and minimal polynomials ⓘ linear changes of variables ⓘ |
| usedFor |
computing Krull dimension
ⓘ
dimension theory in algebraic geometry ⓘ establishing finiteness properties of morphisms of varieties ⓘ proving Hilbert’s Nullstellensatz ⓘ reducing problems on affine varieties to problems on affine space ⓘ relating affine varieties to affine space ⓘ structure theory of finitely generated algebras ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Noether normalization lemma Description of subject: The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
graded Noether normalization
this entity surface form:
Noether's normalization lemma
this entity surface form:
Zariski’s lemma
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
→
relatedTo
→
Noether normalization lemma
ⓘ
subject surface form:
Noether’s AF+BG theorem
this entity surface form:
Noether’s normalization lemma
“Introduction to Commutative Algebra” (with Ian G. Macdonald)
→
hasSubject
→
Noether normalization lemma
ⓘ
subject surface form:
Introduction to Commutative Algebra
this entity surface form:
Noether normalization