Enumerating orders in number fields

J\"urgen Kl\"uners; Markus Kirschmer

REVIEW 2 major objections 3 minor 1 cited by

Enumerating orders in number fields

T0 review · 2 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A tree structure arranges all p-power orders in a number field so that each order of bounded index is enumerated exactly once.

desk verdict A genuinely useful algorithm for enumerating orders in number fields, with a new tree structure that makes the no-duplicate claim credible; worth a serious referee despite minor proof slips and no shipped code. read the letter →

arxiv 2411.08568 v1 pith:D5ENX6QW submitted 2024-11-13 math.NT

Markus Kirschmer , Jürgen Klüners This is my paper · ORCID

classification math.NT MSC 11R5411Y40

keywords ordersinnumberfieldsp-radicaltreeenumerationmultiplierringDade-Taussky-Zassenhaustheoremoverordersalgorithmscomputationaltheory

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

The paper arranges all orders in a number field whose index in the maximal order is a power of a fixed prime p into a tree. The parent of an order is obtained by taking the multiplier ring of the (n-1)st power of its p-radical; a theorem of Dade, Taussky and Zassenhaus ensures this power is invertible, which makes the tree honest. Walking this tree from the root lists every such order exactly once, so all orders of index dividing p^e can be enumerated without the duplicates that plague earlier approaches. The same tree, restricted to a subtree, enumerates the orders above a given order, and the authors' implementation outperforms existing algorithms on examples with larger index.

What carries the argument

The load-bearing tool is the theorem of Dade, Taussky and Zassenhaus ([1, Theorem C]): for every full lattice I in K, some power I^r with 1 ≤ r < n is invertible over the multiplier ring M(I^r). This theorem justifies defining the parent of O as M(J_p(O)^{n-1}), and it makes the set of orders of p-power index a tree with root Z_K. Supporting machinery includes a classification of p-radicals and of subrings of products of finite fields (Propositions 2.11 and 3.2, Algorithm 3.3), subspace enumeration algorithms (Algorithms 4.14 and 4.16), and, for ramified primes, Fröhlich's invertibility criterion and a MeatAxe-based enumeration of submodules (Algorithm 5.4).

What would settle it

Find a full lattice I in a number field K for which none of the powers I, $I^{2}$, ..., $I^{{n-1}}$ is invertible over its multiplier ring; if such a lattice exists, Theorem 4.2 is false and the parent map is not well-defined. A computational check is to run Algorithm 4.3 for a small field and verify that no order is produced twice; a duplicate output would contradict the claimed tree property.

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Extended reading notes

Core claim

The central discovery is that orders of p-power index can be organized into a rooted tree whose root is the maximal order Z_K, with a successor map that is provably acyclic. Specifically, for an order O with p-radical J, the parent is defined as the multiplier ring M($J^{{n-1}}$); by Theorem 4.2, valid for every full lattice, there is an exponent 1 ≤ r < n for which J^r is invertible over this ring, so the definition is legitimate and the relation is a tree. This turns the enumeration of all orders of index dividing p^e into a tree traversal: at each order O, the children correspond to lattices satisfying condition (4.1), which in the unramified case are described by sublattices H of O' with $H^{{n-1}}$=O'. The paper works out this description explicitly and gives a correspondingly concrete algorithm (Algorithm 4.3) that never produces the same order twice.

Load-bearing premise

The tree construction relies on the quoted theorem that for every full lattice I, some power I^r with r<n is invertible over its multiplier ring; the paper does not prove this theorem.

Editorial extensions

If this is right

All orders with index dividing p^e can be output with no duplication, because the tree property makes the history of each order unique.
The overorders of a fixed order Λ are obtained by constraining the same tree walk, which gives large speedups in the examples once the index [Z_K:Λ] is not tiny.
Since the tree is independent of the bound e, the enumeration can be stopped and resumed at any depth, allowing on-demand generation of orders by increasing index.
For unramified p, children of O are enumerated via lattices H with H^{n-1}=O', a condition that can be checked with linear algebra and avoids walking over all subspaces of O/pO.

Reading between the lines

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

The same tree could be used to sample random orders of bounded index uniformly, since each order appears exactly once in the traversal.
The parent map might extend to orders in étale algebras with multiple components, where p-radicals and multiplier rings still behave locally, although the paper only treats number fields.
Because every order appears once, the enumeration is embarrassingly parallel across branches of the tree, so the algorithm could scale to much larger indices on distributed hardware.
For a fixed order Λ, the subtree of orders containing Λ is connected, so the restricted enumeration could be seen as a natural filtration of the lattice of overorders of Λ.

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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

The short version: this is a solid algorithmic number theory paper. The central idea is new and it works. The authors arrange p-power orders in a tree by making the parent of an order O be M(J_p(O)^(n-1)), then reduce enumeration of successors to lattices H satisfying H^r = O. That avoids the two pain points of the older maximal-suborder/minimal-overorder approaches: the same order is not found twice, and most steps do not require walking over all subspaces of F_p-vector spaces. I checked the first example's count of 11 manually and the tree/no-duplicate claim seems right; the stress-test note confirms the parent is uniquely determined and each radical is generated exactly once. The classification of orders with a fixed p-radical via subrings of a product of fields (Theorem 3.1, Proposition 3.2, Corollary 3.4) is also well done and the counting formula is consistent. What deserves credit: the paper gives real proofs, corrects a genuinely faulty lemma from Halter-Koch (Example 5.6), and reports substantial speedups over Hofmann-Sircana on nontrivial examples. The reliance on Dade-Taussky-Zassenhaus is not a hidden assumption; it is a cited, proved external theorem, and the paper uses it appropriately. The ramified case is harder, but the arguments there are supplied in reasonable detail. Soft spots, in proportion: no code is shipped, so reproducibility depends on the Hecke implementation and external pieces like MeatAxe and orbit enumeration. A couple of textual slips in the proof of Algorithm 4.22 (the assertion O = H^(n-1) need not hold when the minimal exponent is smaller, and the n-1 version of Theorem 4.2 is used without derivation) are minor and do not affect the algorithm. The paper could be clearer about why the tree property is stated without an explicit proof, but the missing reasoning is short. Who this is for: people who actually need to compute overorders, ideal class monoids, or bounded-index order lists in number fields. It is a strong candidate for peer review, not because it is flashy but because the method is likely to be used and cited. I would send it to a referee, perhaps asking that referee to check the ramified-case algorithm carefully and to demand a few clarifying sentences in the proof of Algorithm 4.22.

Referee Report

2 major / 3 minor

Summary. The paper introduces a tree on the set of orders in a number field K whose index in the maximal order Z_K is a power of a fixed prime p. The proposed parent of an order O is the multiplier ring M(J_p(O)^{n-1}), and the Dade--Taussky--Zassenhaus theorem is used to guarantee that some power of every p-radical is invertible. Algorithm 4.3 enumerates all orders of index dividing p^e by maintaining a list of p-radicals I; for each I it computes all orders with p-radical I and then all new radicals satisfying condition (4.1). Sections 3--5 supply the local enumeration tools: orders with a given p-radical are classified through subrings of a product of finite fields (Theorem 3.1, Proposition 3.2, Algorithm 3.3); the unramified case is reduced to enumerating lattices H with 1 in H, H^r=O and pH^2 in H (Algorithms 4.18, 4.20, 4.22); the ramified case is handled through invertible ideals and the algorithms of Section 5. Section 6 gives counts and timings for degree-5 fields, including completely split, inert, and completely ramified primes, and compares with the algorithm of Hofmann and Sircana.

Significance. If the missing proof of uniqueness is supplied, this is a strong contribution to computational algebraic number theory. The tree structure removes duplicate enumeration, and the algorithmic core avoids brute-force search over all subspaces: the unramified routine works with vector-space complements and makes essential use of the p-radical. Corollary 3.4 gives an explicit closed counting formula for the number of orders with a fixed p-radical, and the examples give concrete, checkable counts; for instance, Example 6.1(1) reports 11 orders of index 5, and the split/inert/ramified comparison in Example 6.3 is nontrivial. The paper ships an implementation in Hecke, compares timings on the same examples as the previous state of the art, and derives its counts without fitted parameters, which makes the finite output independently verifiable.

major comments (2)

[Section 4, Eq. (4.1)] The sentence after (4.1) states that condition (4.1) arranges the orders of p-power index in Z_K into a tree and hence that Algorithm 4.3 never finds the same order twice. This is the load-bearing correctness property of the enumeration, but it is asserted without proof. Please add a lemma showing: (i) if O' has p-radical J, then J satisfies (4.1) for O=M(J^{n-1}), using Theorem 4.2 together with the fact that M(J^r)=M(J^{n-1}) when J^r is invertible; and (ii) if J satisfies (4.1) for some O, then O=M(J^{n-1}), so the parent is uniquely determined. These assertions are true, but the manuscript should contain the argument rather than the one-sentence 'Note'.
[Section 6] The abstract promises enumeration of orders over a given order Lambda, but Section 6 only sketches the adaptation ('replace Z by (1/p)J_p(Lambda) in Algorithms 4.18 and 4.22') and gives no formal correctness proof for the overorder version. Either state and prove a precise theorem for the overorder enumeration, or explicitly present that part as an implementation with experimental support rather than as part of the paper's central claim.

minor comments (3)

[Section 4, Algorithm 4.22] In the proof of Algorithm 4.22, the statement that O=H^{n-1} follows from (4.2) is not correct in general, since (4.2) only gives H^r=O for some r at most n-1. The needed conclusion p^{n-2}O is contained in H still follows from Remark 4.6 because p^{r-1}O is contained in H and r-1 is at most n-2; the proof should be rewritten with this argument.
[Section 1 and Abstract] There are several typos that should be corrected: 'Th is' in the abstract, 'orderorders' near the end of the first section, and 'in paricular' in reference [1].
[Example 6.1(1)] The same symbol O_i is used both for the set of orders with index at most 5^i and for a particular chain of orders constructed in the example; this is confusing and should be renamed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the tree enumeration derives orders from external theorems and subroutine algorithms, with no fitted inputs or self-referential predictions.

full rationale

The derivation is self-contained against known external results. The tree is defined by making the parent of an order O' equal to M(J_p(O')^{n-1}), and Algorithm 4.3 generates children by solving conditions (4.1); the set of orders being enumerated is never used as an input to produce itself. The uniqueness/no-duplicate assertion is asserted rather than fully proved in the paper, but it follows from Lemma 4.4(1) and Theorem 4.2 without importing the conclusion: if I satisfies (4.1) for O, then IO is invertible and M(I^{n-1}) = O, so each radical is generated from exactly one parent. The quoted Dade-Taussky-Zassenhaus theorem is an external 1962 result, not a result of this paper, and is used as a premise rather than as a disguised restatement of the enumeration. The only self-citation is [7, Lemma 4.3], used for generating units in Algorithm 5.10; that is an independently published subroutine and is not load-bearing for the central tree claim. No constants are fitted, no quantity is renamed as a prediction, and all examples are computed outputs rather than inputs. The minor textual slips noted by the skeptic (use of n-1 rather than the stated r < n, and an auxiliary equality in the proof of Algorithm 4.22) are correctness or exposition issues, not circularity. The central claim is therefore not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rest on standard theorems from the literature rather than on any fitted parameters or invented objects. The deepest external input is the Dade-Taussky-Zassenhaus theorem, followed by Ferrand-Olivier's classification of minimal algebras and Frohlich's invertibility criterion. The computational steps also assume correctness of external computer algebra components (MeatAxe, orbit enumeration). No free parameters and no invented entities are introduced.

assumptions (5)

standard math For every full lattice I in K there exists 1 ≤ r < n with I^r invertible over M(I^r) (Dade-Taussky-Zassenhaus).
Theorem 4.2, cited from [1, Theorem C], is invoked to define the parent M(J_p(O)^(n-1)) and to justify condition (4.1).
standard math Classification of minimal algebras over a field (Ferrand-Olivier): the only minimal algebras are prime-degree field extensions, k × k, and k[X]/(X^2).
Used in Corollary 2.7 and Proposition 2.8 to classify maximal suborders.
standard math Frohlich's invertibility criterion: an ideal a of order O is invertible iff [ZKa:a] = [ZK:O].
Theorem 5.2, used in the ramified case and Algorithm 5.4.
standard math Lemma 5.7's statement that if p ≥ s then H_p contains a unit of O_p^*, proved via [1, Lemma 2.2.7].
Used in Algorithm 5.10's proof to handle the ramified case when p ≥ s.
domain assumption Computational correctness of the MeatAxe and of the orbit enumeration method in [7, Lemma 4.3].
Algorithm 5.4 relies on MeatAxe [8] for enumerating O-submodules, and Algorithm 5.10 uses orbit enumeration with generating set from [7, Lemma 4.3]; these are assumed correct.

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Pith. "Pith review of Enumerating orders in number fields." pith.science (2026). https://pith.science/paper/D5ENX6QW

@misc{pith2026241108568,
  author       = {Pith},
  title        = {Pith review of: Enumerating orders in number fields},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5ENX6QW}},
  note         = {Machine review of arXiv:2411.08568}
}

read the original abstract

We arrange the orders in an algebraic number field in a tree. This tree can be used to enumerate all orders of bounded index in the maximal order as well as the orders over some given order.

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Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

[1]

E. C. Dade, O. Taussky, and H. Zassenhaus. On the theory of orders, in paricular on the semigroup of ideal classes and genera of an order in an algebr aic number ﬁeld. Math. Ann. , 148:31–64, 1962

work page 1962
[2]

Ferrand and J.-P

D. Ferrand and J.-P. Olivier. Homomorphisms minimaux d’ anneaux. J. Algebra, 16:461–471, 1970

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Fieker, W

C. Fieker, W. Hart, T. Hofmann, and F. Johansson. Nemo/He cke: Computer Algebra and Number Theory Packages for the Julia Programming Language. In Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Co mputation, ISSAC ’17, pages 157–164, New York, NY, USA, 2017. ACM

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A. Fr¨ ohlich. Invariants for modules over commutative s eparable orders. Quart. J. Math. Ox- ford Ser. (2) , 16:193–232, 1965

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Halter-Koch

F. Halter-Koch. Ideal semigroups of noetherian domains and ponizovski decompositions. J. Pure Appl. Algebra , 209:763–770, 2007

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Hofmann and C

T. Hofmann and C. Sircana. On the computation of overorde rs. Int. J. Number Theory , 16(4):857–879, 2020

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Kl¨ uners and S

J. Kl¨ uners and S. Pauli. Computing residue class rings and Picard groups of orders. J. Algebra, 292(1):47–64, 2005

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K. Lux, J. M¨ uller, and M. Ringe. Peakword condensation a nd submodule lattices: an appli- cation of the meat-axe. J. Symbolic Comput. , 17(6):529–544, 1994. 22 MARKUS KIRSCHMER AND J ¨URGEN KL ¨UNERS

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S. Marseglia. Computing the ideal class monoid of an orde r. J. Lond. Math. Soc. (2) , 101(3):984–1007, 2020

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S. Marseglia. Local isomorphism classes of fractional ideals of orders in ´ etale algebras, 2023. https://arxiv.org/abs/2311.18571. Universit¨at Bielefeld, Postfach 100131, 33501 Bielefeld, Germany Email address : markus.kirschmer@math.uni-bielefeld.de Universit¨at Paderborn, ...

2023 arXiv

Pith tools

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

[1] [1]

E. C. Dade, O. Taussky, and H. Zassenhaus. On the theory of orders, in paricular on the semigroup of ideal classes and genera of an order in an algebr aic number ﬁeld. Math. Ann. , 148:31–64, 1962

work page 1962

[2] [2]

Ferrand and J.-P

D. Ferrand and J.-P. Olivier. Homomorphisms minimaux d’ anneaux. J. Algebra, 16:461–471, 1970

work page 1970

[3] [3]

Fieker, W

C. Fieker, W. Hart, T. Hofmann, and F. Johansson. Nemo/He cke: Computer Algebra and Number Theory Packages for the Julia Programming Language. In Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Co mputation, ISSAC ’17, pages 157–164, New York, NY, USA, 2017. ACM

work page 2017

[4] [4]

Fr¨ ohlich

A. Fr¨ ohlich. Invariants for modules over commutative s eparable orders. Quart. J. Math. Ox- ford Ser. (2) , 16:193–232, 1965

work page 1965

[5] [5]

Halter-Koch

F. Halter-Koch. Ideal semigroups of noetherian domains and ponizovski decompositions. J. Pure Appl. Algebra , 209:763–770, 2007

work page 2007

[6] [6]

Hofmann and C

T. Hofmann and C. Sircana. On the computation of overorde rs. Int. J. Number Theory , 16(4):857–879, 2020

work page 2020

[7] [7]

Kl¨ uners and S

J. Kl¨ uners and S. Pauli. Computing residue class rings and Picard groups of orders. J. Algebra, 292(1):47–64, 2005

work page 2005

[8] [8]

K. Lux, J. M¨ uller, and M. Ringe. Peakword condensation a nd submodule lattices: an appli- cation of the meat-axe. J. Symbolic Comput. , 17(6):529–544, 1994. 22 MARKUS KIRSCHMER AND J ¨URGEN KL ¨UNERS

work page 1994