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.
Aliases (9)
- Bogomolov multiplier ×1
- Bogomolov used the unramified Brauer group to produce counterexamples ×1
- Noether's problem for abelian groups ×1
- Noether's problem for cyclic groups ×1
- Noether's problem for dihedral groups ×1
- Noether's problem for p-groups ×1
- Noether's problem for symmetric groups ×1
- Noether's problem over the rational numbers ×1
- Voskresenskii studied Noether's problem via algebraic tori ×1
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
→
Noether's problem for cyclic groups → Noether's problem for dihedral groups → Noether's problem for p-groups → Noether's problem for symmetric groups → Noether's problem over algebraically closed fields → Noether's problem over the rational numbers → |
| involves |
finite group actions on fields
→
fixed fields of group actions → rational function fields → |
| knownResult |
Bogomolov used the unramified Brauer group to produce counterexamples
→
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 → 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
→
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
→
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) → |
Referenced by (10)
| Subject (surface form when different) | Predicate |
|---|---|
|
Noether's problem
("Noether's problem for abelian groups")
→
Noether's problem ("Noether's problem for cyclic groups") → Noether's problem ("Noether's problem over the rational numbers") → Noether's problem ("Noether's problem for p-groups") → Noether's problem ("Noether's problem for symmetric groups") → Noether's problem ("Noether's problem for dihedral groups") → |
hasSpecialCase |
|
Noether's problem
("Voskresenskii studied Noether's problem via algebraic tori")
→
Noether's problem ("Bogomolov used the unramified Brauer group to produce counterexamples") → |
knownResult |
|
Emmy Noether
→
|
notableWork |
|
Noether's problem
("Bogomolov multiplier")
→
|
relatedConcept |