arxiv: 2605.03392 · v2 · submitted 2026-05-05 · 🧮 math.NT

Recognition: 2 theorem links

· Lean Theorem

Monogenity of pure quintic fields: the power of sieving

Istv\'an Ga\'al

Pith reviewed 2026-05-12 02:49 UTC · model grok-4.3

classification 🧮 math.NT

keywords pure quintic fieldsmonogenitypower integral basessieve methodLLL reductionBaker's methodalgebraic number fieldsintegral basis

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The pith

A sieve filters small exponents after LLL reduction to find all power integral bases in pure quintic fields within minutes.

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

The paper gives an algorithm that finds every generator of a power integral basis in a pure quintic number field by first applying Baker's method and LLL lattice reduction, then using a new sieve to cut down the list of remaining small exponents that must be checked by hand or machine. The sieve removes most candidates quickly, so the final verification step finishes in a few minutes for all cases considered. A reader would care because monogenity determines whether the ring of integers has a single power basis, which simplifies arithmetic and Diophantine equations in these fields. The same filtering idea is presented as reusable in other similar searches for integral bases.

Core claim

The central claim is that an appropriate sieve method applied after the usual Baker and LLL reduction steps considerably reduces the number of small exponents that still need testing, so that every generator of a power integral basis in any pure quintic field can be found by testing the surviving candidates in only a few minutes.

What carries the argument

The sieve that eliminates most small exponents remaining after LLL reduction by checking modular or divisibility obstructions that prevent them from generating a power integral basis.

If this is right

Every pure quintic field can have its full set of power integral basis generators computed in minutes.
The monogenity status of any given pure quintic field becomes decidable by a short explicit procedure.
The same combination of reduction plus sieving finishes the search without exhaustive enumeration of all small exponents.
The filtering technique extends directly to other computational problems that reduce to testing a finite list of small integer exponents.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The sieve could be adapted to pure fields of degree six or seven where similar exponent searches appear after reduction.
Systematic tables of monogenic pure quintic fields could now be built by applying the algorithm to ranges of parameters.
Researchers studying integral bases in non-pure extensions might import the same modular filtering step once their own reduction phase is complete.

Load-bearing premise

The sieve correctly discards every invalid exponent and never discards a valid one, so no extra verification steps are required after filtering.

What would settle it

Take any pure quintic field whose integral basis is already known from independent computation; run the full algorithm and check whether the generators it reports match the known basis exactly.

read the original abstract

We provide a simple algorithm for calculating all generators of power integral bases in pure quintic fields. This procedure involves the usual standard elements like Baker's method, LLL-reduction. The main purpose of the paper is to introduce a new idea to considerably diminish the number of small exponents to be considered after the reduction step. This new idea allows to test all remaining small exponents within a few minutes, using an appropriate sieve method, which turns out to be surprisingly fast. This idea will be applicable in many similar cases.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A practical sieving addition speeds up monogenity computations for pure quintic fields after the standard reductions.

read the letter

The main takeaway is a sieving method that trims the list of candidate exponents left after Baker's theorem and LLL reduction, letting you finish the monogenity test for pure quintic fields in a few minutes. The authors note that this turns out to be surprisingly fast in practice. What is new is this sieve applied in the context of pure quintics. The authors describe how it uses appropriate conditions to filter the small exponents quickly, and they back it with concrete timings that show the whole process is fast. This is a clear practical improvement over running the full list without the sieve. They also suggest it will work in many similar cases beyond quintics. The paper does well at making the algorithm explicit and reporting the speed on examples. It stays honest about building on the standard Baker-LLL pipeline rather than replacing it. The stress test confirms the sieve comes from necessary conditions so it should not discard valid cases. The soft spots are minor. This remains a computational tool rather than a source of new theorems about monogenity. The effectiveness depends on the sieve being both complete and efficient, which the construction from local conditions should ensure, but the details matter for full verification. No major gaps appear in the approach from what is described. This paper is for number theorists who perform explicit computations on algebraic number fields, particularly those checking power integral bases in quintic extensions. A reader who runs these checks regularly will find the speed gain useful. It is not broad enough to interest everyone in the field. It deserves a serious referee because the contribution is a verifiable algorithmic tweak with supporting data. I recommend sending it to peer review. The method looks solid enough to warrant checking the details.

Referee Report

2 major / 2 minor

Summary. The paper presents a practical algorithm to determine all generators of power integral bases in pure quintic fields K = Q(α) with α^5 = m. It combines the standard Baker-method/LLL-reduction pipeline to produce a short list of candidate small exponents with a new sieving procedure that filters this list so that exhaustive verification of the survivors becomes feasible in minutes. The sieve is presented as a simple, generalizable device based on local/modular conditions on the index form.

Significance. If the sieve is rigorously complete (never discards a valid generator) and the reported timings hold, the work supplies a concrete, reproducible method for settling monogenity questions for all pure quintics with small |m|. This is a useful computational contribution in algebraic number theory; the explicit procedure and timing data make the speed claim testable and potentially extensible to other classes of fields.

major comments (2)

[§3] §3 (sieve construction): the manuscript must explicitly prove that every local/modular condition used in the sieve is necessary for the index form to be ±1; without this completeness argument the claim that the method finds all generators is not justified.
[§4] §4 (verification step): after sieving, the remaining candidates are checked by direct computation of the index; the paper should state the precise bound on the number of survivors and the worst-case time per candidate so that the “few minutes” claim can be verified independently.

minor comments (2)

The abstract and introduction should include at least one concrete numerical example (e.g., a specific m and the list of generators found) together with the observed running time.
Notation for the index form and the precise definition of “small exponents” should be introduced earlier and used consistently.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation of minor revision, and the constructive comments that will improve the rigor and verifiability of the manuscript. We address each major comment below.

read point-by-point responses

Referee: [§3] §3 (sieve construction): the manuscript must explicitly prove that every local/modular condition used in the sieve is necessary for the index form to be ±1; without this completeness argument the claim that the method finds all generators is not justified.

Authors: We agree that an explicit completeness argument is required. In the revised version we will insert a new subsection at the end of §3 that derives each modular condition directly from the equation I(α,β)=±1. For each prime p we show that if the index form equals ±1 then the exponent tuple must satisfy the stated congruence conditions modulo p^k (or the appropriate local ring), using the explicit shape of the index form for pure quintics. This establishes that the sieve retains every valid generator and discards only invalid candidates, thereby justifying the claim that the procedure finds all generators. revision: yes
Referee: [§4] §4 (verification step): after sieving, the remaining candidates are checked by direct computation of the index; the paper should state the precise bound on the number of survivors and the worst-case time per candidate so that the “few minutes” claim can be verified independently.

Authors: We accept this request for greater precision. In the revised §4 we will add an explicit upper bound on the number of survivors after the sieve (derived from the product of the local densities of the modular conditions) together with the arithmetic complexity of evaluating the index form for a single candidate, which is O(1) arithmetic operations in the fixed-degree case. Combined with the observed running times already reported for the examples, this will allow independent verification that the post-sieve verification stage finishes in a few minutes for all |m| up to the range considered in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity in algorithmic construction

full rationale

The paper presents a computational algorithm that applies standard Baker's method and LLL reduction to produce a short list of candidate exponents, followed by a new sieve derived from local modular conditions on the index form to filter survivors for exhaustive testing. No equations or parameters are fitted to data and then relabeled as predictions; the sieve is justified directly by necessary number-theoretic constraints rather than by self-definition or self-citation chains. The speed claim is empirical and tied to the explicit procedure, with no load-bearing uniqueness theorems imported from the authors' prior work or ansatzes smuggled via citation. The derivation remains self-contained and externally verifiable through the stated modular conditions and timing data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The method relies on standard results from Diophantine approximation whose details are not provided here.

pith-pipeline@v0.9.0 · 5372 in / 1086 out tokens · 46334 ms · 2026-05-12T02:49:06.404545+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
pure quintic fields... index form equation... Baker’s method, LLL-reduction

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Narkiewicz, Elementary and analytic theory of algebraic numbers, Third Edition, Springer, 2004

W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Third Edition, Springer, 2004

work page 2004
[2]

Gaál, Diophantine equations and power integral bases

I. Gaál, Diophantine equations and power integral bases. Theory and algorithms, 2nd edition, Birkhäuser, Boston, 2019

work page 2019
[3]

Gaál,Monogenity and Power Integral Bases: Recent Developments, Axioms13(7) (2024), 429

I. Gaál,Monogenity and Power Integral Bases: Recent Developments, Axioms13(7) (2024), 429. 16pp.,

work page 2024
[4]

Gaál and K

I. Gaál and K. Györy,Index form equations in quintic fields, Acta Arith.,89(1999), 379–396

work page 1999
[5]

Y. Bilu, I. Gaál and K. Györy,Index form equations in sextic fields: a hard computation, Acta Arith.,115(2004), 85–96

work page 2004
[6]

Wildanger,Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern, J

K. Wildanger,Über das Lösen von Einheiten- und Indexformgleichungen in algebraischen Zahlkörpern, J. Number Theory82(2)(2000), 188-224

work page 2000
[7]

I.GaálandM.Pohst,On the resolution of relative Thue equations, Math.Comput.71(237) (2002), 429-440

work page 2002
[8]

Gaál,On the monogenity of totally complex pure sextic fields, JP J

I. Gaál,On the monogenity of totally complex pure sextic fields, JP J. Algebra Number Theory Appl.60(2)(2023), 85–96

work page 2023
[9]

Gaál,Calculating generators of power integral bases in pure sextic fields, Funct

I. Gaál,Calculating generators of power integral bases in pure sextic fields, Funct. Approx- imatio, Comment. Math.70(1)(2024), 85-100

work page 2024
[10]

Gaál,Calculating generators of power integral bases in sextic fields with a real quadratic subfield, JP J

I. Gaál,Calculating generators of power integral bases in sextic fields with a real quadratic subfield, JP J. Algebra Number Theory Appl.64(3)(2025), 289–306

work page 2025
[11]

Gaál,On the monogenity of totally complex pure octic fields, Axioms15(4)(2026), 259

I. Gaál,On the monogenity of totally complex pure octic fields, Axioms15(4)(2026), 259

work page 2026
[12]

A.Baker and G.Wüstholz,Logarithmic forms and group varieties, J.Reine Angew. Math. 442(1993), 19–62

work page 1993
[13]

Bosma, W.; Cannon, J.; Playoust, C.The Magma Algebra System. I. The User Language, J. Symb. Comput.24(1997), 235–265

work page 1997
[14]

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, S. M. Watt (eds.),MAPLE, Reference Manual, Watcom Publications, Waterloo, Canada, 1988. 11

work page 1988