Rank of P\'olya Groups in Lecacheux Parametric Family of Quintic Fields

Nimish Kumar Mahapatra; Prem Prakash Pandey

arxiv: 2607.00675 · v1 · pith:UPLZ6XSGnew · submitted 2026-07-01 · 🧮 math.NT

Rank of P\'olya Groups in Lecacheux Parametric Family of Quintic Fields

Nimish Kumar Mahapatra , Prem Prakash Pandey This is my paper

Pith reviewed 2026-07-02 07:15 UTC · model grok-4.3

classification 🧮 math.NT

keywords Pólya groupsquintic fieldsLecacheux family5-rankclass field towerselementary abelian groupsdensity argumentsnon-monogenic fields

0 comments

The pith

In the Lecacheux family of quintic fields the associated Pólya groups are elementary abelian 5-groups that can have arbitrarily large rank.

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

The paper examines the Pólya groups of a one-parameter family of quintic number fields defined by Lecacheux polynomials. It proves that these groups are elementary abelian 5-groups whose 5-rank can be made arbitrarily large by choosing the odd integer parameter appropriately. Density arguments on the parameter show that, for any fixed positive integer k, the set of parameters yielding 5-rank at least k has positive density inside the family. The same density statement, combined with the Golod-Shafarevich theorem, implies that a positive proportion of the fields possess an infinite 5-class field tower. The paper also derives an upper bound on the Pólya number in terms of the order of the Pólya group and exhibits members of the family that are non-monogenic even though their index is one.

Core claim

The 5-rank of the Pólya group attached to a Lecacheux quintic field can be made arbitrarily large; for every positive integer k the subset of odd integers s for which this rank is at least k has positive density, and therefore a positive proportion of the fields admit an infinite 5-class field tower.

What carries the argument

The Pólya group of the number field, which records the obstruction to the existence of a Pólya basis for its ring of integers.

If this is right

For every positive integer k the set of Lecacheux quintic fields whose Pólya group has 5-rank at least k has positive density.
A positive proportion of Lecacheux quintic fields admit an infinite 5-class field tower.
The Pólya number of each field in the family is bounded above by a function of the order of its Pólya group.
Infinitely many members of the family are non-monogenic despite having index one.

Where Pith is reading between the lines

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

Similar density-of-high-rank phenomena may appear in other parametric families of number fields once an analogous Pólya-group computation is available.
The non-monogenic examples with index one suggest that the Pólya group may detect ramification or integral-basis phenomena that the usual index alone misses.
The existence of infinite 5-class field towers for a positive-density subset supplies an infinite supply of fields whose class-group 5-part is unbounded in a controlled way.

Load-bearing premise

The density arguments that produce a positive-density set of parameters for each fixed rank k must remain valid independently of finer arithmetic features of the individual Lecacheux polynomials.

What would settle it

An explicit computation, for a large finite collection of odd integers s, showing that the proportion of fields whose Pólya group has 5-rank at least k tends to zero as k grows, or a single concrete s for which the 5-rank is strictly smaller than the density claim predicts.

read the original abstract

In this article, we study the P\'olya group of a new family of quintic fields, namely Lecacheux quintic fields. We show that the associated P\'olya groups can be arbitrarily large elementary abelian $5$-groups. Using density arguments, we prove that for every positive integer $k$, the set of odd integers $s$ such that the $5-$rank of the P\'olya group of the corresponding Lecacheux quintic field is at least $k$ has a positive density. Combining this with a result of Golod and Shafarevich, we see that for a positive proportion of $s$, the corresponding Lecacheux quintic fields admit an infinte $5-$class field tower. We also establish an upper bound for the P\'olya numbers of these fields in terms of the orders of their corresponding P\'olya groups. In addition, we prove that several fields in this family are non-monogenic despite having index one.

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

The Lecacheux family yields new explicit quintics where 5-rank of the Pólya group is arbitrarily large with positive density, giving positive proportion with infinite 5-towers via Golod-Shafarevich.

read the letter

The main result is that the Lecacheux parametric quintics have Pólya groups that are elementary abelian 5-groups of any desired rank, and that for each k the odd parameters s producing rank at least k form a positive-density set. Combining this with Golod-Shafarevich immediately gives a positive proportion of the fields with infinite 5-class field towers.

The paper applies standard density and splitting arguments to this specific family and derives the necessary conditions on s from the discriminant and ramification in the Lecacheux polynomial. It also supplies an upper bound on the Pólya number in terms of the group order and produces examples that are non-monogenic despite index one. These are concrete additions to the existing literature on Pólya groups.

The density step is the load-bearing piece, exactly as the stress-test note flags. The argument stands or falls on whether the arithmetic conditions forcing high 5-rank remain positive density when restricted to odd s. The abstract states that the calculation works for this family, so the concern does not automatically invalidate the claim; one would simply verify the independence of the congruences in the full proof. Nothing indicates circularity or self-referential definitions.

The citation pattern is appropriate and the methods are the usual ones for this corner of algebraic number theory. No internal contradictions appear from the stated results.

This is for specialists working on class groups, Pólya invariants, and p-class field towers in degree-five fields. A reader looking for new parametric examples with controlled ranks will find usable content.

Send it to peer review. The claims are specific enough to be checked and the family is new.

Referee Report

2 major / 2 minor

Summary. The paper examines the Pólya groups of the Lecacheux parametric family of quintic fields K_s. It proves that these groups can realize arbitrarily large elementary abelian 5-groups, that for every positive integer k the set of odd integers s for which the 5-rank of the Pólya group is at least k has positive (natural) density, and therefore that a positive proportion of these fields have infinite 5-class-field towers by the Golod–Shafarevich theorem. Additional results include an upper bound on the Pólya number in terms of the order of the Pólya group and examples of non-monogenic fields with index 1.

Significance. If the density statements hold, the work supplies an explicit infinite parametric family of quintic fields in which the 5-rank of a distinguished subgroup of the class group is unbounded and in which infinite 5-towers occur with positive density. This gives concrete, computable examples supporting the Golod–Shafarevich mechanism in degree 5 and adds to the limited stock of families with proven unbounded Pólya-group rank. The bound on Pólya numbers and the non-monogenic examples are also of independent interest for the arithmetic of the family.

major comments (2)

[§5] §5 (density arguments): The claim that the set of odd s with 5-rank of the Pólya group ≥ k has positive density for every k is load-bearing for the infinite-tower conclusion. The argument must explicitly verify that the splitting or congruence conditions derived from the discriminant of the Lecacheux polynomial remain independent when restricted to odd s and that their common density is strictly positive; without a quantitative lower bound or an application of the Chinese Remainder Theorem with an explicit modulus, the density could collapse to zero.
[§3] §3, relation between Pólya group and class group: The paper invokes Golod–Shafarevich on the 5-rank of the Pólya group. It must be stated precisely whether the Pólya group is a quotient or a subgroup of the class group and whether the 5-rank lower bound transfers directly to the class-group 5-rank used in the Golod–Shafarevich criterion; any kernel or cokernel of order divisible by 5 would affect the applicability.

minor comments (2)

[Abstract] Abstract, last sentence: “infinte” should be “infinite”.
Notation for the parameter s and the field K_s should be introduced once and used consistently; the current text occasionally switches between “Lecacheux quintic field” and “corresponding field” without cross-reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We respond point by point to the major comments and indicate the changes we will make in revision.

read point-by-point responses

Referee: [§5] §5 (density arguments): The claim that the set of odd s with 5-rank of the Pólya group ≥ k has positive density for every k is load-bearing for the infinite-tower conclusion. The argument must explicitly verify that the splitting or congruence conditions derived from the discriminant of the Lecacheux polynomial remain independent when restricted to odd s and that their common density is strictly positive; without a quantitative lower bound or an application of the Chinese Remainder Theorem with an explicit modulus, the density could collapse to zero.

Authors: We agree that the density claim requires an explicit verification to be fully rigorous. The relevant conditions on the parameter s are congruence conditions modulo powers of 5 (arising from the splitting in the discriminant of the Lecacheux polynomial) together with conditions at a finite set of auxiliary primes; all such moduli are odd. The set of odd integers therefore intersects the arithmetic progressions in a set of positive density, and the Chinese Remainder Theorem yields that the joint density equals the product of the local densities, each of which is strictly positive. In the revised manuscript we will add an explicit statement of the modulus, the independence via CRT, and a quantitative lower bound for the density. revision: yes
Referee: [§3] §3, relation between Pólya group and class group: The paper invokes Golod–Shafarevich on the 5-rank of the Pólya group. It must be stated precisely whether the Pólya group is a quotient or a subgroup of the class group and whether the 5-rank lower bound transfers directly to the class-group 5-rank used in the Golod–Shafarevich criterion; any kernel or cokernel of order divisible by 5 would affect the applicability.

Authors: By definition the Pólya group Po(K) is the subgroup of the class group Cl_K generated by the classes of the prime ideals lying above the primes that divide the discriminant of the ring of integers of K. It is therefore a subgroup, not a quotient. A lower bound on the 5-rank of Po(K) immediately supplies the same lower bound on the 5-rank of Cl_K, which is the quantity required by the Golod–Shafarevich criterion. There is consequently no kernel or cokernel of 5-power order that could interfere. We will insert a precise paragraph in the revised §3 making this subgroup relation and the direct transfer explicit. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses external Golod-Shafarevich theorem and independent density arguments on arithmetic conditions

full rationale

The paper establishes that 5-ranks of Pólya groups in the Lecacheux family can be arbitrarily large by deriving explicit arithmetic conditions on the parameter s (from the discriminant and splitting behavior of the defining polynomial) that force high rank, then shows via standard Dirichlet density that the set of odd s satisfying those conditions for any fixed k has positive density. This is combined with the external Golod-Shafarevich theorem to conclude positive proportion of fields have infinite 5-class field towers. No step reduces by definition, by fitting, or by self-citation chain to its own inputs; the density proof is self-contained against external number-theoretic benchmarks and does not invoke prior results by the same authors as load-bearing. Upper bounds on Pólya numbers and non-monogenic examples are likewise independent of the density claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters or invented entities; the work relies on standard background theorems whose details cannot be audited here.

axioms (2)

standard math Golod-Shafarevich theorem on infinitude of p-class field towers when p-rank is sufficiently large
Invoked in the abstract to conclude infinite towers from high 5-rank.
standard math Existence and basic properties of Pólya groups in number fields
Background assumption for studying the Pólya group in the family.

pith-pipeline@v0.9.1-grok · 5708 in / 1504 out tokens · 32732 ms · 2026-07-02T07:15:27.980136+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 1 canonical work pages

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H. Zantema, Integer valued polynomials over a number field,Manuscripta Math.40, (1982), 155–203. (Nimish Kumar Mahapatra)Indian Institute of Science Education and Research, Thiruvananthapuram, India. Email address:nimish@iisertvm.ac.in (Prem Prakash Pandey)Indian Institute of Science Education and Research, Berham- pur, India. Email address:premp@iiserbpr.ac.in

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[1] [1]

P. J. Cohen, J. L. Chabert, Integer-Valued Polynomials,Math. Surveys Monogr., Amer. Math. Soc., Providence, 1997

1997

[2] [2]

J. -L. Chabert, Factorial groups and P´ olya groups in Galoisian extensions ofQ,Commutative Ring Theory and Applications, Lect. Notes Pure Appl. Math.231, Dekker, New York, (2003), 77–86

2003

[3] [3]

Chabert, From P´ olya fields to P´ olya groups (I) Galois extensions,J

J.-L. Chabert, From P´ olya fields to P´ olya groups (I) Galois extensions,J. Number Theory 203(2019), 360–375

2019

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A. C. Cojocaru and M. R. Murty,An Introduction to Sieve Methods and Their Applications, London Mathematical Society Student Texts, No. 66, Cambridge University Press, 2005

2005

[5] [5]

Gillibert and A

J. Gillibert and A. Levin, A Geometric Approach to Large Class Groups: A Survey, In: Chakraborty, K., Hoque, A., Pandey, P. (eds) Class Groups of Number Fields and Related Topics. Springer, Singapore. https://doi.org/10.1007/978-981-15-1514-9 1

work page doi:10.1007/978-981-15-1514-9

[6] [6]

E. S. Golod, I. R. Shafarevich, On the class field tower,Izv. Akad. Nauk SSSR Ser. Mat.28 (1964), 261–272

1964

[7] [7]

M. -N. Gras, Non monog´ en´ eit´ e de l’anneau des entires des extensions cycliques deQde degr´ e premierl≥5,J. Number Theory23(1986), 347–353

1986

[8] [8]

Heidaryan and A

B. Heidaryan and A. Rajaei, Biquadratic P´ olya fields with only one quadratic P´ olya subfield, J. Number Theory143(2014), 279–285

2014

[9] [9]

Heidaryan and A

B. Heidaryan and A. Rajaei, Some non-P´ olya biquadratic fields with low ramification,Rev. Mat. Iberoam33(2017), 1037–1044

2017

[10] [10]

Hooley, On the power free values of polynomials,Mathematika14(1967), 21–26

C. Hooley, On the power free values of polynomials,Mathematika14(1967), 21–26

1967

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Ishida, The Genus Fields of Algebraic Number Fields,Springer(1976)

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1976

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Lecacheux, Unit´ es d’une famille de corps li´ e’s ´ a la courbeX1(25),Ann

O. Lecacheux, Unit´ es d’une famille de corps li´ e’s ´ a la courbeX1(25),Ann. Inst. Fourier (Grenoble)40(1990), 237–253

1990

[13] [13]

Leriche, Cubic, quartic and sextic P´ olya fields,J

A. Leriche, Cubic, quartic and sextic P´ olya fields,J. Number Theory133(2013), 59–71

2013

[14] [14]

Leriche, About the embedding of a number field in a P´ olya field,J

A. Leriche, About the embedding of a number field in a P´ olya field,J. Number Theory145, (2014), 210–229

2014

[15] [15]

Maarefparvar, P´ olya group in some real biquadratic fields,J

A. Maarefparvar, P´ olya group in some real biquadratic fields,J. Number Theory228, (2021), 1–7

2021

[16] [16]

N. K. Mahapatra and P. P. Pandey, Non-P´ oya fields with large P´ olya groups arising from Lecacheux quintics,Bull. Aust. Math. Soc.110(3)(2024), 468–479

2024

[17] [17]

D. W. Masser, Polynomial Bounds for Diophantine Equations,Amer. Math. Monthly93, (1986), 486–488

1986

[18] [18]

A. K. Silvester, B. K. Spearman, and K. S. Williams, The Conductor of Lecacheux parametric family of cyclic quintic fields,Indian J. pure appl. Math.38(4) (2007), 231–240

2007

[19] [19]

C. W. -W. Tougma, Some questions on biquadric P´ olya fields,J. Number Theory229,(2021), 386–398

2021

[20] [20]

Number Fields with Large Minimal Index

Z. Wolske, “Number Fields with Large Minimal Index”, thesis, University of Toronto, 2018

2018

[21] [21]

von Zylinski, Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer ¨Korper,Math

E. von Zylinski, Zur Theorie der außerwesentlichen Diskriminantenteiler algebraischer ¨Korper,Math. Ann.73(2), (1913), 273–274

1913

[22] [22]

Zantema, Integer valued polynomials over a number field,Manuscripta Math.40, (1982), 155–203

H. Zantema, Integer valued polynomials over a number field,Manuscripta Math.40, (1982), 155–203. (Nimish Kumar Mahapatra)Indian Institute of Science Education and Research, Thiruvananthapuram, India. Email address:nimish@iisertvm.ac.in (Prem Prakash Pandey)Indian Institute of Science Education and Research, Berham- pur, India. Email address:premp@iiserbpr.ac.in

1982