Noether normalization lemma

E29377

mathematical theorem result in algebraic geometry result in commutative algebra

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.

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All labels observed (6)

Label	Occurrences
Noether normalization lemma canonical	6
Noether normalization	2
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.
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
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 ⓘ graded Noether normalization ⓘ linked to: Noether normalization lemma 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

Referenced by (12)

Full triples — surface form annotated when it differs from this entity's canonical label.

Emmy Noether → notableWork → Noether normalization lemma ⓘ

Emmy Noether → knownFor → Noether normalization lemma ⓘ

subject linked to: Emmy

Noether normalization lemma → hasVariant → graded Noether normalization ⓘ