Noether's problem: rationality of the invariant fields of finite permutation groups (the open core of Noether's 1917 paper 'Gleichungen mit vorgeschriebener Gruppe')

The source paper (Emmy Noether, 'Gleichungen mit vorgeschriebener Gruppe', Mathematische Annalen 78 (1917) 221-229) reduces the parameter representation of all equations with a prescribed finite Galois group $G$ to the existence of a 'minimal basis' for the invariant field of $G$ acting by permutation on the variables of $k(x_1,\dots,x_n)$: algebraically independent generators making the fixed field rational (purely transcendental) over the base field $k$. Noether verified this for all subgroups of $S_4$ and left the general case open; that case is the problem the modern literature calls Noether's problem, standardly posed for the regular action, $k(G) = k(x_g \mid g \in G)^G$. The naive positive expectation is refuted: over $\mathbb{Q}$ the cyclic groups $C_{47}$, $C_{113}$, $C_{233}$ (Swan 1969; Voskresenskii 1970 independently found $C_{47}$ and six further prime-order counterexamples) and $C_8$ are not rational, and over algebraically closed fields Saltman (1984) and Bogomolov (1988) constructed $p$-groups that are not even retract rational, detected by the unramified Brauer group and the Bogomolov multiplier $B_0(G)$. The abelian case is completely settled (Lenstra 1974; Plans 2017 completed the cyclic case over $\mathbb{Q}$: $\mathbb{Q}(C_p)$ is rational exactly for $p \le 43$ and $p \in \{61,67,71\}$). Over algebraically closed fields, $p$-groups are settled completely through orders 32 and 243 (order 243 completed by Hoshi-Kang-Yamasaki 2020, who proved the five $\Phi_7$ groups non-rational via degree-3 unramified cohomology), and at order 64 up to one explicitly named exceptional family. The classification for general finite groups remains open with a precise nonempty core: the alternating groups $A_n$ for $n \ge 6$ over any field ($A_5$ was settled by Maeda), the five $\Phi_{13}$ groups of order 64, the order-128 groups with $B_0(G)=0$, and essentially all non-abelian groups over $\mathbb{Q}$. No complete obstruction theory is known: vanishing of all presently computable unramified invariants does not imply rationality.

Background

The inverse Galois problem asks whether every finite group occurs as the Galois group of a field extension of the rational numbers. A century-old strategy for it, formulated in Emmy Noether's 1917 paper 'Gleichungen mit vorgeschriebener Gruppe' ('Equations with prescribed group'), aims at something stronger for each fixed group: a parameter representation of the totality of equations with that prescribed Galois group. Fix a base field $k$ (classically a number field) and embed the finite group $G$ into the symmetric group $S_n$, so that $G$ permutes the variables of the rational function field $k(x_1,\dots,x_n)$ (the field of all fractions of polynomials in the variables; for $S_n$ acting on $n$ variables this models the general degree-$n$ equation). The invariant field or fixed field $k(x_1,\dots,x_n)^G$ is the subfield of rational functions left unchanged by every permutation in $G$. For $G = S_n$, the fundamental theorem of symmetric functions identifies this field with $k(s_1,\dots,s_n)$, generated by the elementary symmetric functions: a purely transcendental extension, meaning one generated by algebraically independent elements. Noether's paper proves that if the invariant field of $G$ admits such generators - her term is a minimal basis, algebraically independent $u_1,\dots,u_n$ with

$$k(x_1,\dots,x_n)^G = k(u_1,\dots,u_n),$$

a property the literature calls rationality (or $k$-rationality) of the invariant field - then the coefficients of the equations with group $G$ acquire a complete parametrization by $n$ free parameters; specializing those parameters over a Hilbertian field (one, such as $k = \mathbb{Q}$, over which Hilbert's irreducibility theorem holds, so that irreducibility of the generic equation survives infinitely many specializations) then realizes $G$ as a Galois group over that field. Noether verified the minimal-basis condition for all subgroups of $S_4$, that is, for all permutation groups of degrees 3 and 4.

For one variable the necessary analogue is Lüroth's theorem (1875): every field between $k$ and $k(x)$ is rational; for two variables over an algebraically closed field of characteristic zero the same holds by a theorem of Castelnuovo (1894). Both are cited in the source paper's reference apparatus and underlie the degree-3 and degree-4 verifications. In three or more variables the analogue is false: the apparatus of the same paper already records a 1912 counterexample of F. Enriques - an explicit non-rational involution of three-dimensional space ('Sopra una involuzione non razionale dello spazio', Rendiconti Lincei (5) 21, no. 1, 81-83) - showing that a field of rational functions in more than two indeterminates need not admit a minimal basis; in modern language, the quotient of a rational variety by a finite group of birational transformations need not be rational. The question the paper therefore leaves open - for which groups does the invariant field admit a minimal basis? - is genuine, and it is the problem the modern literature calls Noether's problem. Its standard modern form lets $G$ act on itself by left translation, $h \cdot x_g = x_{hg}$, and asks whether $k(G) := k(x_g \mid g \in G)^G$ is rational over $k$.

Several weakenings of rationality organize what is known. A field $K$ over $k$ is stably $k$-rational if $K(y_1,\dots,y_m)$ becomes $k$-rational after adjoining finitely many indeterminates, and retract $k$-rational if $K$ is a retract, in the sense of $k$-algebras (the target of an idempotent $k$-algebra homomorphism), of a localization of a polynomial ring over $k$; rationality implies stable rationality implies retract rationality, and both implications are strict. A standard lemma (the no-name lemma) shows that the stable-rationality class of $k(x_1,\dots,x_n)^G$ is independent of the chosen faithful permutation action of $G$, and a theorem of Saltman identifies retract rationality of $k(G)$ with the existence of a generic $G$-Galois extension over $k$ - the modern descendant of Noether's parametrization goal, sufficient for inverse-Galois and local-to-global applications though weaker than rationality. The main obstruction tool is the unramified Brauer group $\mathrm{Br}_{\mathrm{nr}}(K)$: the subgroup of the Brauer group of $K$ (equivalence classes of finite-dimensional central simple algebras over $K$, with cyclic algebras $(L/K,\sigma,a)$ as the basic construction) consisting of classes ramified at no discrete valuation of $K$; it is a stable birational invariant, trivial for rational fields, and Bogomolov's formula identifies it, for $k$ algebraically closed of characteristic not dividing $|G|$, with the Bogomolov multiplier $B_0(G)$: the kernel of the restriction from the second cohomology group $H^2(G,\mathbb{Q}/\mathbb{Z})$ (the Schur multiplier of $G$) to the direct sum of the groups $H^2(A,\mathbb{Q}/\mathbb{Z})$ over all bicyclic subgroups $A$ (abelian subgroups generated by at most two elements). Thus $B_0(G) \ne 0$ proves that $k(G)$ is not even retract rational; but the converse fails, and vanishing of all presently known unramified invariants does not imply rationality, so no complete decision procedure exists.

What is known: for abelian groups, rationality holds over fields containing enough roots of unity (a 1915 theorem of E. Fischer, recorded in Noether's own reference apparatus), and Lenstra (1974) gave a complete criterion for abelian groups over an arbitrary field in terms of cyclotomic arithmetic; Plans (2017) completed the cyclic case over $\mathbb{Q}$, where $\mathbb{Q}(C_p)$ is rational exactly for $p \le 43$ and $p \in \{61,67,71\}$. For cyclic groups of prime order the arithmetic behaviour is governed by cyclotomic norms: rationality of $\mathbb{Q}(C_p)$ requires the existence of an element of norm $\pm p$ in the ring of integers of the $(p-1)$-st cyclotomic field, so non-existence of such an element implies non-rationality. Negative answers exist in abundance: over $\mathbb{Q}$, $\mathbb{Q}(C_{47})$ is not rational (Swan 1969; independently Voskresenskii 1970) and neither is $\mathbb{Q}(C_8)$; over an algebraically closed field, Saltman (1984) and Bogomolov (1988) constructed $p$-groups (of orders $p^9$ and $p^6$) that are not retract rational. Positively, over algebraically closed fields (or fields containing enough roots of unity): all $p$-groups of order at most $p^4$ are rational (Chu-Kang 2001); all groups of order 32 are rational (Chu-Hu-Kang-Prokhorov 2008); among the groups of order 64, the nine with nonzero Bogomolov multiplier (the isoclinism family $\Phi_{16}$) are not even retract rational, and all the others are rational over quadratically closed fields except possibly five groups of the isoclinism family $\Phi_{13}$ (Chu-Hu-Kang-Kunyavskii 2010, who also computed $B_0$ exactly for order 64); and among the groups of order 243, all groups outside the isoclinism families $\Phi_7$ and $\Phi_{10}$ are rational over fields containing enough roots of unity, while the three $\Phi_{10}$ groups have nonzero Bogomolov multiplier and the five $\Phi_7$ groups have nonzero degree-3 unramified cohomology, so none of these eight is rational (Chu-Hoshi-Hu-Kang 2015; Hoshi-Kang-Yamasaki 2020). Two groups are isoclinic when their commutator structures match (isomorphisms between the central quotients and the commutator subgroups compatible with the commutator map); isoclinic $p$-groups share the same Bogomolov multiplier, which is why the exceptional cases organize into isoclinism families $\Phi$. The problem is thus neither open in the naive sense (the answer is known to be negative in general) nor solved: the classification, and in particular each of the open cases listed in the problem statement, remains active - most recently through work on reduction modulo $p$ of the problem for $p$-groups (Ambrosi-Valloni 2025).

Problem Statement

Noether's problem, in the form left open by the source paper. Let $k$ be a field and let $G$ be a finite group acting faithfully on the variables of the rational function field $k(x_1,\dots,x_n)$ by permutation, i.e. embedded as a subgroup of $S_n$; the standard modern form takes the regular action of $G$ on itself, with invariant field $k(G) = k(x_g \mid g \in G)^G$. Determine for which pairs $(G \hookrightarrow S_n, k)$ - a finite group together with a fixed faithful permutation representation and a base field - the invariant field is rational over $k$ - that is, admits a minimal basis in Noether's sense: algebraically independent elements $u_1,\dots,u_m$ with $k(x_1,\dots,x_n)^G = k(u_1,\dots,u_m)$.

Scope: the target is the classification over the two standard settings $k = \mathbb{Q}$ and $k = \mathbb{C}$ (and over general $k$ where a uniform answer is possible) for arbitrary finite groups, with rationality itself - not merely stable or retract rationality - as the target property, and retaining the original generality (no restriction to solvable, nilpotent, or small groups). The abelian case is settled (Lenstra), as is the cyclic case over $\mathbb{Q}$ (Plans), and negative answers exist in general; the concrete open frontier to be resolved includes the following independently answerable cases, while the full classification remains the overarching objective: (i) the alternating groups $A_n$ for all $n \ge 6$, open over any base field in the regular-action form $k(A_n)$, with $A_5$ settled affirmatively by Maeda; (ii) the five isoclinism-family $\Phi_{13}$ groups of order 64 over quadratically closed fields (the 2020 survey poses the related rationality problem for an explicitly described fixed field as its Question 3.23); (iii) the groups of order 128 with vanishing Bogomolov multiplier; and (iv) over $\mathbb{Q}$, any single non-abelian group beyond the scattered families settled to date.

References

1. Emmy Noether, 'Gleichungen mit vorgeschriebener Gruppe', Mathematische Annalen 78 (1917) 221-229, DOI 10.1007/BF01457099, https://doi.org/10.1007/BF01457099 - the source paper.
2. J. Lüroth, 'Beweis eines Satzes über rationale Curven', Mathematische Annalen 9 (1875) 163-165, DOI 10.1007/BF01443371, https://doi.org/10.1007/BF01443371 - Lüroth's theorem (the one-indeterminate case); cited in the source paper's apparatus.
3. G. Castelnuovo, 'Sulla razionalità delle involuzioni piane', Mathematische Annalen 44 (1894) 125-155, DOI 10.1007/BF01446977, https://doi.org/10.1007/BF01446977 - the two-indeterminate case over algebraically closed fields of characteristic zero; cited in the source paper's apparatus.
4. Federigo Enriques, 'Sopra una involuzione non razionale dello spazio', Rendiconti dell'Accademia dei Lincei, Serie V, vol. 21, no. 1 (session of 21 January 1912), pp. 81-83, zbMATH Open JFM 43.0702.01, https://zbmath.org/2626760 - Enriques's counterexample: an explicit non-rational involution of three-dimensional space, showing that a field of rational functions in more than two indeterminates need not admit a minimal basis. Cited in the source paper's apparatus as 'Rend. Acc. Linc., Vol. 21, 21. Jan. 1912' (without title there).
5. E. Fischer, 'Die Isomorphie der Invariantenkörper der endlichen Abelschen Gruppen linearer Transformationen', Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-physikalische Klasse (1915), pp. 77-80, zbMATH Open JFM 45.0200.01, https://zbmath.org/2615557, digitized at the Göttingen Digitalization Center: http://resolver.sub.uni-goettingen.de/purl?GDZPPN002504103 - Fischer's theorem: the invariant field of a finite abelian group of exponent e is rational over any field containing the e-th roots of unity (of characteristic not dividing e). Cited in the source paper's apparatus ('Gött. Nachr. 1915').
6. Richard G. Swan, 'Invariant rational functions and a problem of Steenrod', Inventiones Mathematicae 7 (1969) 148-158, DOI 10.1007/BF01389798, https://doi.org/10.1007/BF01389798 - first counterexamples to Noether's problem over $\mathbb{Q}$ (cyclic groups of orders 47, 113, 233).
7. V. E. Voskresenskii, 'On the question of the structure of the subfield of invariants of a cyclic group of automorphisms of the field Q(x_1,...,x_n)', Mathematics of the USSR-Izvestiya 4 (1970) 371-380, DOI 10.1070/IM1970v004n02ABEH000910, https://doi.org/10.1070/im1970v004n02abeh000910 - independent cyclic counterexamples (orders 47, 167, 359, 383, 479, 503, 719).
8. H. W. Lenstra Jr., 'Rational functions invariant under a finite abelian group', Inventiones Mathematicae 25 (1974) 299-325, DOI 10.1007/BF01389732, https://doi.org/10.1007/BF01389732 - complete criterion for abelian groups over an arbitrary field.
9. David J. Saltman, 'Generic Galois extensions and problems in field theory', Advances in Mathematics 43 (1982) 250-283, DOI 10.1016/0001-8708(82)90036-6, https://doi.org/10.1016/0001-8708(82)90036-6 - retract rationality and generic Galois extensions.
10. David J. Saltman, 'Noether's problem over an algebraically closed field', Inventiones Mathematicae 77 (1984) 71-84, DOI 10.1007/BF01389135, https://doi.org/10.1007/BF01389135 - meta-abelian $p$-groups of order $p^9$ that are not retract rational over algebraically closed fields.
11. F. A. Bogomolov, 'The Brauer group of quotient spaces by linear group actions', Mathematics of the USSR-Izvestiya 30 (1988) 455-485, DOI 10.1070/IM1988v030n03ABEH001024, https://doi.org/10.1070/im1988v030n03abeh001024 - counterexamples of order $p^6$ and the formula identifying the unramified Brauer group of the invariant field with the Bogomolov multiplier.
12. Takashi Maeda, 'Noether's problem for A_5', Journal of Algebra 125 (1989) 418-430, DOI 10.1016/0021-8693(89)90174-9, https://doi.org/10.1016/0021-8693(89)90174-9 - $k(A_5)$ is rational over any field.
13. Richard G. Swan, 'Noether's Problem in Galois Theory', in: Emmy Noether in Bryn Mawr, Springer, New York, 1983, pp. 21-40, DOI 10.1007/978-1-4612-5547-5_2, https://doi.org/10.1007/978-1-4612-5547-5_2 - early survey of the problem's status.
14. Huah Chu, Ming-chang Kang, 'Rationality of p-group actions', Journal of Algebra 237 (2001) 673-690, DOI 10.1006/jabr.2000.8615, https://doi.org/10.1006/jabr.2000.8615 - $p$-groups of order at most $p^4$.
15. Ming-Chang Kang, 'Noether's problem for metacyclic p-groups', Advances in Mathematics 203 (2006) 554-567, DOI 10.1016/j.aim.2005.05.004, https://doi.org/10.1016/j.aim.2005.05.004 - positive results for metacyclic $p$-groups.
16. Huah Chu, Shou-Jen Hu, Ming-chang Kang, Y. G. Prokhorov, 'Noether's problem for groups of order 32', Journal of Algebra 320 (2008) 3022-3035, DOI 10.1016/j.jalgebra.2008.07.007, https://doi.org/10.1016/j.jalgebra.2008.07.007 - all groups of order 32.
17. H. Chu, S.-J. Hu, M.-c. Kang, B. E. Kunyavskii, 'Noether's Problem and the Unramified Brauer Group for Groups of Order 64', International Mathematics Research Notices 2010 (2010) 2329-2366, DOI 10.1093/imrn/rnp217, https://doi.org/10.1093/imrn/rnp217 - order 64 settled except possibly five $\Phi_{13}$ groups over quadratically closed fields, with $B_0$ computed exactly.
18. Primož Moravec, 'Unramified Brauer groups of finite and infinite groups', American Journal of Mathematics 134 (2012) 1679-1704, DOI 10.1353/ajm.2012.0046, https://doi.org/10.1353/ajm.2012.0046 - algorithmic computation of $B_0$ for all solvable groups of order at most 729 (the orders 512, 576 and 640 excluded from the tabulation); his Table 1 gives 9 of the 267 groups of order 64, 230 of the 2328 groups of order 128, and 3 of the 67 groups of order 243 with $B_0 \ne 0$.
19. Akinari Hoshi, Ming-Chang Kang, Boris E. Kunyavskii, 'Noether's problem and unramified Brauer groups', Asian Journal of Mathematics 17 (2013) 689-714, DOI 10.4310/AJM.2013.v17.n4.a8, https://doi.org/10.4310/ajm.2013.v17.n4.a8 - $B_0$ determined for all groups of order $p^5$ (nonzero precisely on one isoclinism family).
20. Huah Chu, Akinari Hoshi, Shou-Jen Hu, Ming-chang Kang, 'Noether's problem for groups of order 243', Journal of Algebra 442 (2015) 233-259, DOI 10.1016/j.jalgebra.2015.03.010, https://doi.org/10.1016/j.jalgebra.2015.03.010 - order 243: all groups outside the isoclinism families $\Phi_7$ and $\Phi_{10}$ are rational over fields containing enough roots of unity, and the three $\Phi_{10}$ groups are not retract rational (the $\Phi_7$ case completed negatively by reference 28).
21. Bernat Plans, 'On Noether's rationality problem for cyclic groups over Q', Proceedings of the American Mathematical Society 145 (2017) 2407-2409, DOI 10.1090/proc/13438, https://doi.org/10.1090/proc/13438 - completes the cyclic case over $\mathbb{Q}$.
22. Akinari Hoshi, 'Noether's problem and rationality problem for multiplicative invariant fields: a survey', RIMS Kôkyûroku Bessatsu B77 (2020) 29-53, arXiv:2010.01517, https://arxiv.org/abs/2010.01517 - primary state-of-the-art source for the status statements and open cases (its open item for the $\Phi_7$ groups of order 243 was resolved by reference 28).
23. Ming-chang Kang, Jian Zhou, 'Noether's problem for some semidirect products', Advances in Mathematics 368 (2020) 107164, DOI 10.1016/j.aim.2020.107164, https://doi.org/10.1016/j.aim.2020.107164 - positive results for certain semidirect products.
24. Hang Fu, Ming-chang Kang, Baoshan Wang, Jian Zhou, 'Noether's problem for some subgroups of S14: The modular case', Journal of Algebra 568 (2021) 529-546, DOI 10.1016/j.jalgebra.2020.10.019, https://doi.org/10.1016/j.jalgebra.2020.10.019 - positive results in the modular characteristic.
25. Olivier Wittenberg, 'Park City lecture notes: around the inverse Galois problem', arXiv:2302.13719 (v4, 17 October 2025), https://arxiv.org/abs/2302.13719 - source of the standard formulation (Problem 1.4.1) and of the historical attribution of the problem to Noether.
26. Emiliano Ambrosi, Domenico Valloni, 'Reduction modulo p of Noether's problem', Algebra & Number Theory 19 (2025) 2359-2367, DOI 10.2140/ant.2025.19.2359, https://doi.org/10.2140/ant.2025.19.2359 - most recent progress (reduction modulo $p$ for $p$-groups).
27. Urban Jezernik, Primož Moravec, 'Bogomolov Multipliers of Groups of Order 128', Experimental Mathematics 23 (2014), no. 2, 174-180, DOI 10.1080/10586458.2014.886980, https://doi.org/10.1080/10586458.2014.886980 - recomputation of the Bogomolov multipliers of all groups of order 128: exactly 11 isoclinism families ($\Phi_{16}, \Phi_{30}, \Phi_{31}, \Phi_{37}, \Phi_{39}, \Phi_{43}, \Phi_{58}, \Phi_{60}, \Phi_{80}, \Phi_{106}, \Phi_{114}$) have nonzero multiplier, 230 groups in total, so Noether's problem has a negative answer for these.
28. Akinari Hoshi, Ming-chang Kang, Aiichi Yamasaki, 'Degree three unramified cohomology groups and Noether's problem for groups of order 243', Journal of Algebra 544 (2020) 262-301, DOI 10.1016/j.jalgebra.2019.08.008, https://doi.org/10.1016/j.jalgebra.2019.08.008 (arXiv:1710.01958) - completes the classification at order 243: for a group $G$ of order $3^5$, $H^3_{\mathrm{nr}}(\mathbb{C}(G),\mathbb{Q}/\mathbb{Z}) \ne 0$ if and only if $G$ belongs to $\Phi_7$, and $\mathbb{C}(G)$ is rational if and only if $G$ belongs to neither $\Phi_7$ nor $\Phi_{10}$.