Noether's problem

E29549

mathematical problem problem in field theory problem in invariant theory

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.

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All labels observed (12)

Label	Occurrences
Noether's problem canonical	2
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 finite groups	1
Noether's problem for p-groups	1
Noether's problem for symmetric groups	1
Noether's problem over the rational numbers	1
Noether’s problem	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 self-linksurface differs ⓘ surface form: Noether's problem for abelian groups Noether's problem self-linksurface differs ⓘ surface form: Noether's problem for cyclic groups Noether's problem self-linksurface differs ⓘ surface form: Noether's problem for dihedral groups Noether's problem self-linksurface differs ⓘ surface form: Noether's problem for p-groups Noether's problem self-linksurface differs ⓘ surface form: Noether's problem for symmetric groups Noether's problem over algebraically closed fields ⓘ Noether's problem self-linksurface differs ⓘ surface form: Noether's problem over the rational numbers
involves	finite group actions on fields ⓘ fixed fields of group actions ⓘ rational function fields ⓘ
knownResult	Noether's problem self-linksurface differs ⓘ surface form: 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 ⓘ Noether's problem self-linksurface differs ⓘ surface form: 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	Noether's problem self-linksurface differs ⓘ surface form: 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 normalization lemma ⓘ surface form: 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 (13)

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

Emmy Noether → notableWork → Noether's problem ⓘ

Noether's problem → relatedConcept → Noether's problem self-linksurface differs ⓘ

this entity surface form: Bogomolov multiplier