Noether's problem

Noether's problem
Name	Noether's problem
Field	Algebra, Field theory, Invariant theory
Introduced	1913
Introduced by	Emmy Noether
Status	Partially resolved; many open cases

Noether's problem Noether's problem asks whether the fixed field of a permutation action of a finite group on a rational function field is itself rational (purely transcendental) over a base field. The question links classic figures and institutions in algebra such as Emmy Noether, David Hilbert, Emil Artin, Hendrik Lenstra, and Serre-related themes, and it has motivated work at centers like the Mathematics Institute, the Institute for Advanced Study, and universities including Göttingen, Harvard University, and Princeton University.

Statement of the problem

Let G be a finite group acting by permuting coordinates on the rational function field K(x_g : g in G), where K is a base field and x_g are indeterminates indexed by elements of G. The problem asks whether the fixed subfield K(x_g)^G is K-rational, i.e., isomorphic to a purely transcendental extension K(t_1,...,t_n). This formulation connects to classical questions treated by Hilbert's Hilbert's Theorem 90, Noether's own work, and later formulations related to the Inverse Galois Problem, the Brauer group, and the structure of group cohomology.

Historical background and motivation

Emmy Noether posed the question in the context of understanding invariants under finite group actions, building on earlier invariant-theory work by David Hilbert, Arthur Cayley, and James Joseph Sylvester. The problem was motivated by attempts to classify rational invariants for permutation representations appearing in Galois extensions studied by Évariste Galois and formalized in the language of field theory by figures like Emil Artin and Richard Dedekind. Later work by Igor Shafarevich, John Tate, and Jean-Pierre Serre connected the problem to cohomological obstructions and to the behavior of the Brauer group in number-theoretic settings such as those studied at Princeton University and the Courant Institute. Modifications and refinements were influenced by computational advances at institutions like Max Planck Society and collaborations among researchers affiliated with University of Tokyo, Moscow State University, and ETH Zurich.

Known results and criteria

Several positive and negative criteria are known. If G is a cyclic group of prime order and K contains appropriate roots of unity, then classical results of Noether and later refinements by Emmy Noether's successors yield rationality; such cases were clarified by Masayoshi Noether and by investigations influenced by the work of Helmut Hasse. For abelian groups, a criterion due to Swan and later sharpened by Lenstra relates rationality to the vanishing of certain elements in the Brauer group and to the structure of cyclotomic fields studied by Kummer and Leopoldt. Nonabelian negative results began with counterexamples constructed by Swan and later by Saltman, using cohomological obstructions invented by Serre and techniques related to Galois cohomology. For p-groups, significant contributions were made by Kuniyoshi, Endo, and Miyata, while systematic criteria for nilpotent or metacyclic groups were developed by researchers associated with University of Tokyo and Kyoto University.

Examples and counterexamples

Positive examples include permutation actions of cyclic groups under cyclotomic hypotheses explored by Lenstra and classical rationality cases tied to Luroth-type theorems. Counterexamples arise for certain nonabelian groups: Swan provided early examples, while Saltman produced families of groups for which the fixed field is not rational by exhibiting nontrivial unramified Brauer classes. Explicit groups yielding failure of rationality include certain metabelian groups and p-groups studied by Saltman and Bogomolov. The Noether-type problem over finite fields exhibits different behavior, with counterexamples tied to work by Kuniyoshi and computational instances verified by teams at University of Michigan and University of Tokyo.

Techniques and methods of proof

Proofs and obstructions draw on a mix of invariant theory, algebraic geometry, and cohomology. Central tools include the computation of the unramified Brauer group via Bogomolov's multiplier, the use of exact sequences in group cohomology popularized by Jean-Pierre Serre, and the application of specialization techniques reminiscent of Hilbert's irreducibility methods. Techniques from the theory of quadratic forms and Galois cohomology developed by Alexander Grothendieck and Jean-Louis Colliot-Thélène are used alongside explicit construction of generic polynomials tied to the Inverse Galois Problem pursued by researchers at Harvard University and University of Cambridge. Computational approaches leverage algorithms from computer algebra systems originating in efforts at Max Planck Institute and Symbolic computation groups.

Connections and applications

Noether's problem interfaces with the Inverse Galois Problem by relating rationality of parameter spaces to realizability of groups as Galois groups over fields like Q and Q_p. It impacts the study of rationality of algebraic varieties investigated by Shafarevich and Birch and Swinnerton-Dyer contexts, and it informs the construction of generic polynomials important in explicit class field theory and computational number theory communities at ETH Zurich and Max Planck Society. The problem also connects to moduli problems studied at the Institute for Advanced Study and to birational geometry frameworks advanced by Iskovskikh and Manin.

Category:Algebraic number theory