Computing class groups by induction with generalised norm relations

CANARI); Fabrice Etienne (IMB; UB

REVIEW 2 major objections 4 minor 14 references

Computing class groups by induction with generalised norm relations

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Class groups can be computed by reducing to lower-degree subfields via generalized norm relations.

desk verdict Genuinely new norm-relation generalization with impressive examples, but the main proof has a fixable gap and the abstract overstates the GRH qualification. read the letter →

arxiv 2411.13124 v2 pith:5P5U7RC3 submitted 2024-11-20 math.NT

Fabrice Etienne (IMB , UB , CANARI) This is my paper · ORCID

classification math.NT MSC 11R2911R3211Y40

keywords generalizednormrelationsS-unitgroupsclassgroupcomputationMackeyfunctorsGaloisclosurecompositumsHeckealgebrasnumberfields

open problems The Riemann Hypothesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the class group and S-unit group of a number field can be computed from the S-unit groups of smaller subfields whenever a certain algebraic identity, called a generalized norm relation, holds in the Galois group. The identity is a group-algebra equation $N_H = \sum_i a_i N_{J_i} b_i$, and its existence is shown equivalent to a purely field-theoretic condition expressible with compositums. The author proves Theorem A: under the generalized Riemann hypothesis, there is an algorithm, polynomial in the input size, that takes such subfield unit bases and returns a basis of the S-units of the target field, hence its class group. This would matter because the cost of computing class groups by the standard subexponential method grows badly with degree and discriminant, and an inductive reduction to lower-degree fields circumvents that bottleneck. The paper also reports a concrete success: a degree-105 field whose class group was computed in about five days, where classical norm-relation methods did not finish in five months.

What carries the argument

The central objects are generalized norm relations in the group algebra $\mathbb{Q}[G]$: an equality $N_H = \sum_{i=1}^{\ell} a_i N_{J_i} b_i$, where $N_H = \sum_{h\in H} h$ is the norm element of a subgroup $H$ fixing the target field $K$, and the $J_i$ fix the auxiliary subfields. The mechanism that carries the argument is the transfer of this group-algebra identity to cohomological Mackey functors: by proposition 3.5, a relation with $\varphi \circ \psi = d\cdot \mathrm{id}$ between permutation modules gives, for every cohomological Mackey functor $M$, maps between $M(J_i)$ and $M(H)$ whose composition is $d\cdot \mathrm{id}$. Applied to $M(H) = O_{\tilde{K}^H,S}^{\times}$, this produces the S-unit basis. The coefficient bound $c(\mathcal{J},H) \mid |G|^2$ makes the saturation step finite and polynomial.

What would settle it

Take the $C_7 \times A_5$ field of Example 6.2, compute its S-units with Algorithm 4.3 and independently with a direct method; any discrepancy in the returned basis or in the final class group refutes the central claim. A cheaper test: verify the bound $c(\mathcal{J},H) \mid |G|^2$ on a large finite group with a known relation; one counterexample would refute Theorem 2.16 and with it the polynomial complexity.

Watch

Extended reading notes

Core claim

The central claim is that norm relations can be generalised from the trivial subgroup to an arbitrary subgroup $H$ of the Galois group $G$ of the Galois closure. Given subfields $K_i = \tilde{K}^{J_i}$, a generalised norm relation $N_H = \sum_i a_i N_{J_i} b_i$ in $\mathbb{Q}[G]$ transfers to the S-unit groups: the Mackey functor $M(H) = O_{\tilde{K}^H,S}^{\times}$ admits maps $\varphi_M$ and $\psi_M$ with $\varphi_M \circ \psi_M = d\cdot \mathrm{id}$, so a basis of the S-units of the $K_i$ yields a basis of the S-units of $K$. Equivalently, the relation exists iff there is a surjective $\mathbb{Q}[G]$-module morphism from $\bigoplus_i \mathbb{Q}[G/J_i]$ to $\mathbb{Q}[G/H]$, which is checked without computing Galois groups using the compositum action $C\cdot x = N_{C/L}(\iota_K(x))$. The optimal coefficient $c(\mathcal{J},H)$ is proved to divide $|G|^2$, which yields the polynomial bound via saturation over primes dividing $(n!)^2$.

Load-bearing premise

The polynomial-time guarantee rests on the generalized Riemann hypothesis, a standard unproven assumption about the distribution of prime numbers; if that hypothesis fails, the algorithm's run-time bound no longer follows.

Editorial extensions

If this is right

If Theorem A holds, class groups of number fields admitting generalized norm relations can be computed in polynomial time in the input size under GRH, including the S-unit bases of subfields.
For Galois extensions, classical norm relations become a special case, so the method extends the reach of the prior norm-relation framework to non-Galois fields.
Fields with no classical norm relation in any quotient may still admit generalized norm relations; the paper exhibits $S_5$ and $C_7 \times A_5$ examples with concrete speedups.
The optimal-denominator bound drops from $|G|^3$ to $|G|^2$, improving the provable complexity of the saturation step.
A practical variant, correct on termination, verified $\mathrm{Cl}(K)=1$ for a degree-105 field in about 5 days CPU time, where classical methods did not finish in over 5 months.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the polynomial-time claim is right, computing class groups of large-degree fields could be redirected to searching for generalized norm relations in the Galois group rather than attacking the field directly; the author explicitly leaves the search problem open.
The compositum-action criterion suggests a possible algorithmic shortcut: one could try to find relations by linear algebra over $\mathbb{Q}$ on the embeddings, without ever computing the Galois group, which may scale to fields beyond current Galois-closure limits.
Because the transfer works for any cohomological Mackey functor, the same induction could in principle compute other invariants such as unit groups of group rings or K-theory, though the paper only mentions S-units and class groups.
The bound $c(\mathcal{J},H) \mid |G|^2$ hints that the saturation over primes dividing $(n!)^2$ might be replaceable by a smaller set depending only on the group, which would improve constants.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The short version: this paper has a genuinely new idea—generalized norm relations with an arbitrary subgroup H—and it backs it up with impressive computations, including a class group that went from a five-month computation to five days. But the proof of the main theorem has a gap, and the abstract overstates what is proved.

What is actually new: replacing the norm element N_H (for trivial H) by N_H for any subgroup H, and showing that this buys you relations that do not come from classical norm relations in subgroups or quotients (their S4, S5, A5, C7×A5 examples). The compositum criterion (Theorem 2.21) is a useful practical tool, letting you check relations without computing Galois groups. The bound c(J,H) | |G|² improves the |G|³ bound for classical norm relations. The computational examples are the real payoff and look credible.

The soft spots are real but fixable. In the proof of Theorem 4.4, the author asserts that Proposition 2.14 gives φ∘ψ = id. It gives φ∘ψ = c·id. That matters: the induced map on S-unit groups is not necessarily surjective; its image only contains c-th powers. The algorithm's saturation step is clearly designed to handle this, and with the bound c | (n!)², the argument can be made: repeatedly adjoining p-th roots for all p | c turns finite-index inclusion into equality. But the proof as written does not supply that argument, so the correctness claim is unsupported as it stands. A referee should ask for this to be written out.

The second issue is presentation: Theorem 4.4 assumes GRH, while the abstract and Theorem A present the polynomial-time claim unqualified. The algorithm may in fact need GRH only for standard subroutines, but the statement should match the theorem. The paper also does not release code or data, which is a minor reproducibility setback given the computational claims.

On the citation pattern, the paper engages fairly with the prior norm-relation literature and the Mackey functor sources; no red flags there.

Who should read this: anyone doing computational algebraic number theory, especially on class groups of high-degree fields. It deserves a serious referee: the idea is sound, the examples are strong, and the gap is patchable. I would not desk-reject it, but I would insist on a revised proof of Theorem 4.4 and a corrected abstract.

Referee Report

2 major / 4 minor

Summary. The paper introduces 'generalised norm relations' in the group algebra Q[G] for a finite group G, extending the classical norm relations of Biasse--Fieker--Hofmann--Page. It proves equivalent characterisations (Proposition 2.7), a field-theoretic criterion (Theorem 2.21), and a bound on the optimal coefficient (Theorem 2.16). The main algorithmic contribution is Algorithm 4.3, which claims to compute a basis of the S-unit group of a number field K from bases of S-unit groups of subfields K_i, with polynomial-time complexity; Theorem 4.4 states this under GRH. The paper also presents a heuristic class-group algorithm (Algorithm 4.6) and large computational examples, including a degree-105 field whose class group is computed in about 5 CPU days.

Significance. If the correctness gap identified below is repaired and the GRH qualification is stated consistently, this is a useful contribution to computational algebraic number theory. The generalised norm relation framework genuinely extends [2], the bound c(J,H) | |G|^2 improves the analogous |G|^3 bound for classical norm relations, and the examples (e.g. S5, A5, C7 x A5) show that the method can handle fields that are out of reach of direct computation. The paper includes explicit algorithms and reproducible computational experiments, which are valuable even where the complexity proof needs repair.

major comments (2)

[§4, Theorem 4.4 proof; Definition 2.12; Proposition 2.14; Proposition 3.5] The proof of Theorem 4.4 states: 'Since there is a generalised norm relation, we know that there exists an integer c, a surjective morphism ... φ ... and an injective morphism ... ψ ..., such that φ∘ψ = id (by proposition 2.14).' This is not what Proposition 2.14 and Definition 2.12 deliver. They only guarantee φ∘ψ = c(J,H)·id for a positive integer c(J,H), with φ having image of finite index rather than being surjective. Consequently Proposition 3.5 gives φ_M∘ψ_M = c·id_{M(H)}, i.e. c·M(H) ⊆ im(φ_M), and the induced map on S-unit groups is not shown to be surjective. The correctness claim 'this proves the correctness' therefore does not follow as written. The missing step is to invoke Theorem 2.16 (c(J,H) divides |G|^2, and |G| ≤ n!) and to prove explicitly that the p-saturations in step 5 of Algorithm 4.3, performed for all primes dividing (n!)^2 with multiplicity, convert this finite-index inclusion into equality. This argument is absent from the proof. A related slip occurs at the end of the proof of Proposition 2.14, where the text writes 'Ψ ∘ Φ' when Definition 2.12 requires 'Φ ∘ Ψ'; with the correct order the conclusion is φ∘ψ = c·id, still not the identity.
[Abstract, Introduction (Theorem A), §4 Theorem 4.4] There is a statement-level inconsistency about GRH. The abstract and Theorem A advertise 'a polynomial time algorithm' without qualification, while Theorem 4.4 begins 'Assume the generalized Riemann Hypothesis (GRH). Then this algorithm is correct and its complexity is polynomial in the size of the input.' As written, the unconditional polynomial-time claim in Theorem A and the abstract is not supported by the paper's own theorem. The GRH hypothesis must be added to Theorem A and the abstract, or an unconditional proof must be supplied.

minor comments (4)

[Proposition 2.14] The last line of the proof writes 'Ψ ∘ Φ' where the definition of c(J,H) and the surrounding argument require 'Φ ∘ Ψ'; this typo should be corrected to avoid confusion with the identity/c·id issue in Theorem 4.4.
[Theorem 1.18] In the final sentence of the proof, the notation 'N_{C/M}(ι_L(x))' introduces a symbol M that has not been defined; the intended field should be named (apparently the compositum field C considered as an extension of L).
[Abstract and various definitions] There are several typographical slips, e.g. 'so me' in the abstract, 'a set a subgroups' in Definitions 2.2 and 2.4, and the OCR-style rendering 'C4×C4 2' in Example 6.1; these should be cleaned up.
[References] Reference [7] is listed as 'GE.' with an incomplete author name; the full name and details should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the norm-relation input is an independent hypothesis and the S-unit transfer uses external Mackey functor results; the GRH and φ∘ψ = c·id gaps are correctness issues, not circularity.

full rationale

The derivation chain is not circular under any of the seven enumerated patterns. The generalized norm relation is an input condition, not an output: Theorem 2.21 (with Algorithm 4.1) verifies the relation independently by linear algebra over compositum actions, and Theorem 2.11 shows that relations with fields outside the Galois closure can be replaced by relations with subfields inside it. The transfer from group-algebra relations to S-unit groups is made through Proposition 3.5, which is an external Mackey-functor result cited from Boltje ([3, corollary 1.4]), not from the author's prior work. No parameter is fitted to the data being 'predicted': the algorithm takes bases of the subfield S-unit groups as input and computes the target S-unit group by images under compositums followed by p-saturation; the output is not used to define or fit the relation. There is no self-citation chain, no imported uniqueness theorem, and no renaming of a known empirical pattern. The concerns in the skeptic brief are genuine but are not circularity: (1) Theorem 4.4 assumes GRH while the abstract and Theorem A state polynomial time unconditionally, so the unconditional claim is unsupported by the paper's own theorem statement; (2) the proof of Theorem 4.4 asserts 'φ∘ψ = id (by proposition 2.14)' although Definition 2.12 and Proposition 2.14 only yield φ∘ψ = c·id with φ having image of finite index, so the surjectivity of the induced map on S-unit groups is not established as written. These are correctness/completeness gaps in the proof, not reductions of the conclusion to the input by construction. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method introduces no fitted constants. The central claim assumes standard semisimple algebra and Mackey functor facts, a GRH assumption for the complexity theorem, and that the input already contains S-unit bases of the smaller fields. No new particles, fields, or mathematical entities are postulated.

assumptions (4)

standard math The group algebra Q[G] is semisimple and decomposes into simple modules via central idempotents.
Used in Proposition 2.7, 2.13, and 2.16 to convert a module-theoretic surjection into a norm relation.
standard math The transfer theorem for cohomological Mackey functors (Proposition 3.5) holds as stated in [3, corollary 1.4].
This is the bridge that turns a group algebra relation into relations among S-unit groups; the paper cites Boltje [3] and Yoshida [13] rather than proving it.
domain assumption The generalized Riemann Hypothesis.
Theorem 4.4 invokes GRH to prove correctness and polynomial complexity of Algorithm 4.3.
domain assumption The input includes a Z-basis of the S-unit group of each subfield Ki.
Theorem A and Algorithm 4.3 take these bases as given; the algorithm does not compute them.

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Cite this review

Pith. "Pith review of Computing class groups by induction with generalised norm relations." pith.science (2026). https://pith.science/paper/5P5U7RC3

@misc{pith2026241113124,
  author       = {Pith},
  title        = {Pith review of: Computing class groups by induction with generalised norm relations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5P5U7RC3}},
  note         = {Machine review of arXiv:2411.13124}
}

read the original abstract

We introduce a generalisation of norm relations in the group algebra Q[G], where G is a finite group. We give some properties of these relations, and use them to obtain relations between the S-unit groups of different subfields of the same Galois extension of Q, of Galois group G. Then we deduce an algorithm to compute the class groups of some number fields by reducing the problem to fields of lower degree. We compute the class groups of some large number fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages

[2]

Norm relations and computa- tional problems in number ﬁelds

Jean-Fran¸ cois Biasse et al. “Norm relations and computa- tional problems in number ﬁelds”. English. In: J. Lond. Math. Soc., II. Ser. 105.4 (2022), pp. 2373–2414. issn: 0024-

work page 2022
[1]

Computing the residue of the Dedekind zeta function

Karim Belabas and Eduardo Friedman. “Computing the residue of the Dedekind zeta function”. In: Math. Comp. 84.291 (2015), pp. 357–369. issn: 0025-5718,1088-6842. doi: 10.1090/S0025-5718-2014-02843-3 . url: https://doi.org/10.1090/S0025-5718

work page doi:10.1090/s0025-5718-2014-02843-3 2015
[3]

Class group relations from Burnside ring idempotents

Robert Boltje. “Class group relations from Burnside ring idempotents”. English. In: J. Number Theory 66.2 (1997), pp. 291–305. issn: 0022-314X. doi: 10.1006/jnth.1997.2165

work page arXiv 1997
[4]

Beziehungen zwischen Klassenzahlen von Teilk¨ orpern eines galoisschen K¨ orpers

Richard Brauer. “Beziehungen zwischen Klassenzahlen von Teilk¨ orpern eines galoisschen K¨ orpers”. German. In:Math. Nachr. 4 (1951), pp. 158–174. issn: 0025-584X. doi: 10.1002/mana.3210040116

work page doi:10.1002/mana.3210040116 1951
[5]

A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds

Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds . English. S´ emin. Th´ eor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 27-41 (1990). 1990

work page 1990
[6]

Curtis and Irving Reiner

Charles W. Curtis and Irving Reiner. Methods of represen- tation theory with applications to ﬁnite groups and orders. Volume 1. English. Paperback edition. New York etc.: John Wiley &— Sons, 1990. isbn: 0-471-52367-4. 33

work page 1990
[7]

Algorithms related to multiplicative representations of algebraic numbers, PhD thesis

GE. “Algorithms related to multiplicative representations of algebraic numbers, PhD thesis”. In: University of Cali- fornia, Berkeley (1993)

work page 1993
[8]

Asymptotically fast triangularization of matrices over rings

James L. Hafner and Kevin S. McCurley. “Asymptotically fast triangularization of matrices over rings”. English. In: Discrete algorithms. Proceedings of the 1st annual ACM- SIAM symposium, held January 22-24, 1990 in San Fran- cisco, CA (USA) . Philadelphia, PA (USA): SIAM, 1990, pp. 194–200. isbn: 0-89871-251-3

work page 1990

Show all 14 references

[9]

Fac- toring polynomials with rational coeﬃcients

A. K. Lenstra, H. W. jun. Lenstra, and L´ aszl´ o Lov´ asz. “Fac- toring polynomials with rational coeﬃcients”. English. In: Math. Ann. 261 (1982), pp. 515–534. issn: 0025-5831. doi: 10.1007/BF01457454. url: https://eudml.org/doc/182903

1982 doi
[10]

Complexity of lattice problems

Daniele Micciancio and Shaﬁ Goldwasser. Complexity of lattice problems. A cryptographic perspective.English. Vol. 671. Kluwer Int. Ser. Eng. Comput. Sci. Boston, MA: Kluwer Academic Publishers, 2002. isbn: 0-7923-7688-9

2002
[11]

available from http://pari.math.u-bordeaux.fr/

PARI/GP version 2.15.4. available from http://pari.math.u-bordeaux.fr/. The PARI Group. Univ. Bordeaux, 2023

2023
[12]

I. Reiner. Maximal orders. English. Reprint of the 1975 orig- inal. Vol. 28. Lond. Math. Soc. Monogr., New Ser. Oxford: Oxford University Press, 2003. isbn: 0-19-852673-3

1975
[13]

On G-functors. II: Hecke operators and G-functors

Tomoyuki Yoshida. “On G-functors. II: Hecke operators and G-functors”. English. In: J. Math. Soc. Japan 35 (1983), pp. 179–190. issn: 0025-5645. doi: 10.2969/jmsj/03510179. 34

1983
[6107]

doi: 10.1112/jlms.12563

doi

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [2]

Norm relations and computa- tional problems in number ﬁelds

Jean-Fran¸ cois Biasse et al. “Norm relations and computa- tional problems in number ﬁelds”. English. In: J. Lond. Math. Soc., II. Ser. 105.4 (2022), pp. 2373–2414. issn: 0024-

work page 2022

[2] [1]

Computing the residue of the Dedekind zeta function

Karim Belabas and Eduardo Friedman. “Computing the residue of the Dedekind zeta function”. In: Math. Comp. 84.291 (2015), pp. 357–369. issn: 0025-5718,1088-6842. doi: 10.1090/S0025-5718-2014-02843-3 . url: https://doi.org/10.1090/S0025-5718

work page doi:10.1090/s0025-5718-2014-02843-3 2015

[3] [3]

Class group relations from Burnside ring idempotents

Robert Boltje. “Class group relations from Burnside ring idempotents”. English. In: J. Number Theory 66.2 (1997), pp. 291–305. issn: 0022-314X. doi: 10.1006/jnth.1997.2165

work page arXiv 1997

[4] [4]

Beziehungen zwischen Klassenzahlen von Teilk¨ orpern eines galoisschen K¨ orpers

Richard Brauer. “Beziehungen zwischen Klassenzahlen von Teilk¨ orpern eines galoisschen K¨ orpers”. German. In:Math. Nachr. 4 (1951), pp. 158–174. issn: 0025-584X. doi: 10.1002/mana.3210040116

work page doi:10.1002/mana.3210040116 1951

[5] [5]

A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds

Johannes Buchmann. A subexponential algorithm for the determination of class groups and regulators of algebraic number ﬁelds . English. S´ emin. Th´ eor. Nombres, Paris/Fr. 1988-89, Prog. Math. 91, 27-41 (1990). 1990

work page 1990

[6] [6]

Curtis and Irving Reiner

Charles W. Curtis and Irving Reiner. Methods of represen- tation theory with applications to ﬁnite groups and orders. Volume 1. English. Paperback edition. New York etc.: John Wiley &— Sons, 1990. isbn: 0-471-52367-4. 33

work page 1990

[7] [7]

Algorithms related to multiplicative representations of algebraic numbers, PhD thesis

GE. “Algorithms related to multiplicative representations of algebraic numbers, PhD thesis”. In: University of Cali- fornia, Berkeley (1993)

work page 1993

[8] [8]

Asymptotically fast triangularization of matrices over rings

James L. Hafner and Kevin S. McCurley. “Asymptotically fast triangularization of matrices over rings”. English. In: Discrete algorithms. Proceedings of the 1st annual ACM- SIAM symposium, held January 22-24, 1990 in San Fran- cisco, CA (USA) . Philadelphia, PA (USA): SIAM, 1990, pp. 194–200. isbn: 0-89871-251-3

work page 1990