Noether's problem
E29549
Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
All labels observed (12)
How this entity was disambiguated
This entity first appeared as the object of triple T228996 — 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's problem Context triple: [Emmy Noether, notableWork, Noether's problem]
-
A.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
B.
Noether normalization lemma
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.
-
C.
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.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
- 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's problem Target entity description: Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
-
A.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
B.
Noether normalization lemma
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.
-
C.
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.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Dyson’s transform in number theory
Dyson’s transform in number theory is a combinatorial technique introduced by Freeman Dyson to manipulate and relate integer partitions, particularly in the study of partition identities and congruences.
- F. None of above. chosen
Statements (54)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in field theory ⓘ problem in invariant theory ⓘ |
| asksAbout | rationality of fixed fields under finite group actions ⓘ |
| asksWhether |
the fixed field of a finite group acting on a rational function field is purely transcendental
ⓘ
the invariant field k(x_g : g in G)^G is k-rational ⓘ |
| dependsOn |
the base field k
ⓘ
the finite group G ⓘ |
| field |
Galois theory
ⓘ
algebraic geometry ⓘ algebraic number theory ⓘ field theory ⓘ invariant theory ⓘ |
| hasAnswerType | yes-or-no question ⓘ |
| hasSpecialCase |
Noether's problem for abelian groups
ⓘ
linked to:
Noether's problem
Noether's problem for cyclic groups ⓘ
linked to:
Noether's problem
Noether's problem for dihedral groups ⓘ
linked to:
Noether's problem
Noether's problem for p-groups ⓘ
linked to:
Noether's problem
Noether's problem for symmetric groups ⓘ
linked to:
Noether's problem
Noether's problem over algebraically closed fields ⓘ Noether's problem over the rational numbers ⓘ
linked to:
Noether's problem
|
| involves |
finite group actions on fields
ⓘ
fixed fields of group actions ⓘ rational function fields ⓘ |
| knownResult |
Bogomolov used the unramified Brauer group to produce counterexamples
ⓘ
linked to:
Noether's problem
Endo and Miyata obtained positive results for certain abelian groups ⓘ Swan constructed counterexamples over the rational numbers ⓘ Voskresenskii studied Noether's problem via algebraic tori ⓘ
linked to:
Noether's problem
for many finite abelian groups over algebraically closed fields of characteristic zero the answer is yes ⓘ there exist finite groups for which the answer to Noether's problem is no ⓘ |
| motivation |
constructing generic polynomials for finite groups
ⓘ
understanding generic Galois extensions with group G ⓘ |
| namedAfter | Emmy Noether ⓘ |
| questionForm | is k(G) k-rational? ⓘ |
| relatedConcept |
Bogomolov multiplier
ⓘ
linked to:
Noether's problem
Noether field ⓘ essential dimension ⓘ generic Galois extension ⓘ generic polynomial ⓘ invariant field ⓘ purely transcendental extension ⓘ rational field extension ⓘ unramified Brauer group ⓘ versal torsor ⓘ |
| relatedTo |
Noether's normalization lemma
ⓘ
linked to:
Noether normalization lemma
birational geometry of quotient varieties ⓘ inverse Galois problem ⓘ rationality problem for quotient varieties ⓘ |
| standardFormulation | given a field k and a finite group G, is k(x_g : g in G)^G purely transcendental over k? ⓘ |
| status | open in full generality ⓘ |
| timePeriod | formulated in the early 20th century ⓘ |
| typicalNotation | k(G) for the fixed field k(x_g : g in G)^G ⓘ |
| typicalSetup |
a base field k and a finite group G acting by k-automorphisms
ⓘ
a finite group G acting on a rational function field k(x_g : g in G) ⓘ |
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's problem Description of subject: Noether's problem is a fundamental question in invariant theory and field theory that asks whether the fixed field of a finite group acting on a rational function field is itself a purely transcendental (rational) extension.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
linked to: Noether's problem
Noether's problem
→
knownResult
→
Bogomolov used the unramified Brauer group to produce counterexamples
ⓘ
linked to: Noether's problem
linked to: Noether's problem
Swan constructed counterexamples over the rational numbers
→
relatedTo
→
Noether's problem for finite groups
ⓘ
linked to: Noether's problem