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arxiv: 2605.00949 · v1 · submitted 2026-05-01 · 🧮 math.NT

Monogenic Fields from Polynomial Compositions with Applications

Anuj Jakhar , Ravi Kalwaniya , Prabhakar Yadav This is my paper

Pith reviewed 2026-05-09 18:59 UTC · model grok-4.3

classification 🧮 math.NT
keywords monogenic number fieldspolynomial compositionring of integersindex of ordersdiscriminantsasymptotic estimatesirreducible polynomialsalgebraic number theory
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The pith

Necessary and sufficient conditions determine when composed polynomials generate monogenic number fields.

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

The paper establishes exact conditions under which a root alpha of the composed polynomial f_i(x^k + b) generates both the number field K = Q(alpha) and its full ring of integers Z_K. This matters for simplifying the arithmetic of algebraic integers in these specific families of degree-n extensions with n at least 3. The authors derive the conditions for two explicit families of irreducible base polynomials f1 and f2, obtain asymptotic counts of the monogenic cases under natural assumptions on the coefficients, and apply the results to produce polynomials whose discriminants are not square-free while also studying associated differential equations.

Core claim

For the families f1(x) = x^n + c sum_{j=1}^n (a x)^{n-j} and f2(x) = x^n + c sum_{j=1}^n a^{j-1} x^{n-j} that are irreducible over Z of degree n >= 3, the composed polynomials f_i(x^k + b) yield monogenic fields K_i = Q(alpha_i) precisely when the index [Z_{K_i} : Z[alpha_i]] equals 1, which holds under explicit divisibility conditions on a, b, c, and k.

What carries the argument

The index [Z_K : Z[alpha]] for alpha a root of the composed polynomial f_i(x^k + b), determined via necessary and sufficient coefficient conditions that force this index to equal one.

If this is right

  • The number fields K_i generated by roots of the composed polynomials are monogenic exactly when the stated coefficient conditions hold.
  • Asymptotic estimates count the monogenic polynomials inside each family under the natural assumptions on the parameters.
  • Explicit constructions exist of polynomials in these families whose discriminants are not square-free.
  • The monogenic property supplies information about the behavior of solutions to certain differential equations tied to the polynomials.

Where Pith is reading between the lines

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

  • The index criterion could be tested on small-degree examples to verify the boundary between monogenic and non-monogenic cases.
  • The asymptotic counts suggest a positive density of monogenic members within the larger space of all degree-n polynomials of similar shape.
  • The non-square-free discriminant constructions may intersect with problems about square-free values of discriminants in other polynomial families.

Load-bearing premise

The base polynomials f1 and f2 are irreducible over the integers for every degree n at least three.

What would settle it

A concrete triple of integers a, b, c together with k >= 2 such that f1 or f2 is irreducible of degree n >= 3 yet the root alpha of f_i(x^k + b) satisfies [Z_K : Z[alpha]] > 1.

read the original abstract

A number field $K$ is called \emph{monogenic} if its ring of integers $\mathbb{Z}_K$ can be expressed as a simple ring extension $\mathbb{Z}[\alpha]$ for some $\alpha \in \mathbb{Z}_K$. A monic irreducible polynomial $f(x)\in\mathbb{Z}[x]$ is said to be monogenic if one of its roots generates both the number field and its ring of integers. In this article, we establish the necessary and sufficient conditions for $[\mathbb{Z}_{K_i}:\mathbb{Z}[\alpha_i]]=1$, where $K_i=\mathbb{Q}(\alpha_i)$ and $\alpha_i$ is a root of the composed polynomial $f_i(x^k+b)$ for $i=1,2$. Here, $f_1(x)=x^n+c\sum_{j=1}^{n}(ax)^{n-j}\in\mathbb{Z}[x]$ and $f_2(x)=x^n+c\sum_{j=1}^{n}a^{j-1}x^{n-j}\in\mathbb{Z}[x]$ are irreducible polynomials of degree $n\ge 3$. In addition, we derive asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions. As an application of our main results, we construct a class of polynomials with non-square-free discriminants. We also analyze the behavior of solutions to certain related differential equations.

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.

Referee Report

3 major / 2 minor

Summary. The paper claims to establish necessary and sufficient conditions for the index [ℤ_{K_i} : ℤ[α_i]] = 1, where K_i = ℚ(α_i) and α_i is a root of the composed polynomial f_i(x^k + b) for i=1,2, with f_1(x) = x^n + c ∑_{j=1}^n (a x)^{n-j} and f_2(x) = x^n + c ∑_{j=1}^n a^{j-1} x^{n-j} being irreducible polynomials of degree n ≥ 3. It also derives asymptotic estimates for the number of monogenic polynomials in these families under natural assumptions, constructs a class of polynomials with non-square-free discriminants as an application, and analyzes solutions to certain related differential equations.

Significance. If the central claims hold, the explicit necessary and sufficient conditions for monogenicity in these composed families would provide concrete constructions of monogenic number fields, which are of interest in algebraic number theory for controlling the ring of integers and discriminant. The application to non-square-free discriminants could yield useful examples. However, the quantitative asymptotics are weakened by dependence on unspecified assumptions, limiting their strength.

major comments (3)
  1. [Abstract] Abstract and main theorem statement: The necessary and sufficient conditions for [ℤ_{K_i} : ℤ[α_i]] = 1 require that the composed polynomial f_i(x^k + b) be irreducible over ℚ (so that [K_i : ℚ] equals the degree nk of the composition). The manuscript only asserts that the base polynomials f_1 and f_2 are irreducible of degree n ≥ 3; no criterion, proof, or additional assumption is supplied to guarantee irreducibility of the composition. This is load-bearing, as a factorization would alter the field degree and invalidate the stated index conditions.
  2. [Asymptotics section] Asymptotics paragraph: The asymptotic estimates for the number of monogenic polynomials in the families depend on 'natural assumptions' that are not explicitly stated, defined, or justified anywhere in the manuscript. Without these, the quantitative claims cannot be verified or reproduced.
  3. [Applications section] Application to non-square-free discriminants: The construction is presented as following from the monogenicity conditions, but the manuscript does not verify that the resulting discriminants are indeed non-square-free or explain how the index formula directly implies this property without further computation.
minor comments (2)
  1. The analysis of solutions to related differential equations is mentioned in the abstract but appears disconnected from the number-theoretic results; the link should be clarified or the material moved to an appendix if retained.
  2. Notation for the families f_1(x) and f_2(x) would benefit from an explicit low-degree example (e.g., n=3) to illustrate the composition f_i(x^k + b).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and will revise the paper to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract and main theorem statement: The necessary and sufficient conditions for [ℤ_{K_i} : ℤ[α_i]] = 1 require that the composed polynomial f_i(x^k + b) be irreducible over ℚ (so that [K_i : ℚ] equals the degree nk of the composition). The manuscript only asserts that the base polynomials f_1 and f_2 are irreducible of degree n ≥ 3; no criterion, proof, or additional assumption is supplied to guarantee irreducibility of the composition. This is load-bearing, as a factorization would alter the field degree and invalidate the stated index conditions.

    Authors: We agree that irreducibility of the composed polynomial f_i(x^k + b) is necessary for the field degree to be nk and for the index conditions to apply as stated. The manuscript focuses on the monogenicity criteria assuming the composition defines a degree-nk extension, but does not explicitly state or prove this. In revision we will add an explicit hypothesis to the statements of the main theorems that f_i(x^k + b) is irreducible over ℚ, together with a brief remark that this can be checked case-by-case via Eisenstein's criterion or reduction modulo primes for concrete parameter choices. The monogenicity conditions themselves remain valid under this hypothesis. revision: yes

  2. Referee: [Asymptotics section] Asymptotics paragraph: The asymptotic estimates for the number of monogenic polynomials in the families depend on 'natural assumptions' that are not explicitly stated, defined, or justified anywhere in the manuscript. Without these, the quantitative claims cannot be verified or reproduced.

    Authors: The phrase 'natural assumptions' was intended to refer to standard conditions in the literature on counting irreducible monogenic polynomials (e.g., that the parameters a, b, c, k vary over integers with gcd conditions ensuring irreducibility and that n, k ≥ 3 are fixed). We acknowledge that these were not spelled out. In the revised version we will replace the phrase with an explicit list of hypotheses in the asymptotics section, including the required ranges and coprimality conditions on the parameters, and briefly justify why they are needed for the counting arguments. revision: yes

  3. Referee: [Applications section] Application to non-square-free discriminants: The construction is presented as following from the monogenicity conditions, but the manuscript does not verify that the resulting discriminants are indeed non-square-free or explain how the index formula directly implies this property without further computation.

    Authors: The application constructs families where the index is 1 and the discriminant of the composed polynomial is computed via the standard formula involving the index and the discriminant of the minimal polynomial. For suitable parameter choices the discriminant formula visibly contains squared prime factors arising from the composition. We agree that an explicit verification was omitted. In revision we will add a short paragraph with a concrete numerical example (or a general argument) showing that the discriminant is divisible by p² for a prime p dividing b or c, thereby confirming it is non-square-free when the index equals 1. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives necessary and sufficient conditions for the index [Z_{K_i}:Z[alpha_i]]=1 directly from the explicit forms of the composed polynomials f_i(x^k + b) and the given irreducibility of the base polynomials f1, f2 of degree n>=3. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. Asymptotic counts are obtained under explicitly stated natural assumptions using standard enumeration techniques in algebraic number theory. The central claims rest on independent algebraic computations rather than renaming or smuggling prior results. Even if the irreducibility of the compositions requires additional verification (a potential correctness gap), this does not create circularity per the enumerated patterns, as the derivation does not presuppose its own outputs.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from algebraic number theory regarding irreducibility and field extensions, plus unspecified natural assumptions for the counting results.

free parameters (1)
  • natural assumptions parameters
    Used for asymptotic estimates but not specified in the abstract.
axioms (2)
  • domain assumption f1 and f2 are irreducible monic polynomials in Z[x] of degree n >= 3
    Explicitly stated as given for the families.
  • domain assumption The composed polynomial f_i(x^k + b) is considered for parameters k, b such that the root generates the field
    Core to the setup of K_i and alpha_i.

pith-pipeline@v0.9.0 · 5554 in / 1444 out tokens · 53542 ms · 2026-05-09T18:59:56.588835+00:00 · methodology

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    Barman, A

    R. Barman, A. Jakhar, R. Kalwaniya, and P. Yadav, Arithmetic aspects of number fields generated by polynomial families, arXiv preprint arXiv:2602.19726 (2026)

    work page arXiv 2026

  2. [2]

    Barman, A

    R. Barman, A. Narode, and V. W agh, On the monogenity of polynomials with non-squarefree discr iminants, Acta Arith. (2025)

    work page 2025

  3. [3]

    , On monogeneity of reciprocal polynomials , The Ramanujan Journal 69 (2026), 56

    work page 2026

  4. [4]

    , On the discriminant and index of a certain class of polynomia ls, arXiv preprint arXiv:2602.15641 (2026), To appear in Bulletin Australian Math Soc

    work page arXiv 2026

  5. [5]

    Bhargava, A

    M. Bhargava, A. Shankar, and X. W ang, Squarefree values of polynomial discriminants i , Inventiones Mathematicae 228 (2022), 1037–1073

    work page 2022

  6. [6]

    Castillo, A dynamical characterization for monogenity at every level of some infinite 2-towers , Canadian Mathematical Bulletin 65 (2022), 806–814

    M. Castillo, A dynamical characterization for monogenity at every level of some infinite 2-towers , Canadian Mathematical Bulletin 65 (2022), 806–814

    work page 2022

  7. [7]

    Cohen, A course in computational algebraic number theory , Springer-Verlag, Berlin-Heidelberg, 1993

    H. Cohen, A course in computational algebraic number theory , Springer-Verlag, Berlin-Heidelberg, 1993

    work page 1993

  8. [8]

    Cullinan, The discriminant of a composition of two polynomials , https://studylib.net/doc/8187082/the-discriminant-of-a-composition- Accessed: 2025-08-07

    J. Cullinan, The discriminant of a composition of two polynomials , https://studylib.net/doc/8187082/the-discriminant-of-a-composition- Accessed: 2025-08-07

    work page arXiv 2025

  9. [9]

    Dedekind, Über den zusammenhang zwischen der theorie der ideale und de r theorie der höheren kongruenzen , Göttingen Abhandlungen 23 (1878), 1–23

    R. Dedekind, Über den zusammenhang zwischen der theorie der ideale und de r theorie der höheren kongruenzen , Göttingen Abhandlungen 23 (1878), 1–23

    work page

  10. [10]

    4, 416–425

    Paul Erdős, Arithmetical properties of polynomials , Journal of the London Mathematical Society 28 (1953), no. 4, 416–425

    work page 1953

  11. [11]

    Evertse and K

    J. Evertse and K. Győry, Discriminant equations in diophantine number theory , Cambridge University Press, 2016

    work page 2016

  12. [12]

    Gaál, Diophantine equations and power integral bases: Theory and a lgorithms, 2nd ed., Birkhäuser/Springer, Cham, 2019

    I. Gaál, Diophantine equations and power integral bases: Theory and a lgorithms, 2nd ed., Birkhäuser/Springer, Cham, 2019

    work page 2019

  13. [13]

    , On the monogenity of certain binomial compositions , JP Journal of Algebra, Number Theory and Applications 57 (2022), 1–16

    work page 2022

  14. [14]

    Gaál, Monogenity and power integral bases: Recent developments , Axioms 13 (2024), no

    I. Gaál, Monogenity and power integral bases: Recent developments , Axioms 13 (2024), no. 7, 429

    work page 2024

  15. [15]

    Granville, Abc allows us to count squarefrees , International Mathematics Research Notices 19 (1998), 991–1009

    A. Granville, Abc allows us to count squarefrees , International Mathematics Research Notices 19 (1998), 991–1009

    work page 1998

  16. [16]

    Harrington and L

    J. Harrington and L. Jones, Some new polynomial discriminant formulas , Lithuanian Mathematical Journal 61 (2021), 483–490

    work page 2021

  17. [17]

    Jakhar, R

    A. Jakhar, R. Kalwaniya, and P. Yadav, On the monogenity and galois group of certain classes of poly nomials, Mathematica Slovaca 74 (2024), no. 5, 1147–1154

    work page 2024

  18. [18]

    , A study of monogenity of binomial composition , Acta Arithmetica 221 (2025), 387–395

    work page 2025

  19. [19]

    4, 108231

    , A study of monogeneity of trinomial compositions: Qualitat ive and quantitative aspects , Journal of Pure and Applied Algebra 230 (2026), no. 4, 108231

    work page 2026

  20. [20]

    Jones, Monogenic polynomials with non-squarefree discriminant , Proceedings of the American Mathematical Society 148 (2020), no

    L. Jones, Monogenic polynomials with non-squarefree discriminant , Proceedings of the American Mathematical Society 148 (2020), no. 4, 1527–1533

    work page 2020

  21. [21]

    , Some new infinite families of monogenic polynomials with non -squarefree discriminant , Acta Arithmetica 197 (2021), 213–219

    work page 2021

  22. [22]

    , The monogenity of power-compositional eisenstein polynomi als, Annales Mathematicae et Informaticae 55 (2022), 93–113

    work page 2022

  23. [23]

    , On the monogenicity of power-compositional shanks polynom ials, Funct. Approx. Comment. Math. 69 (2023), 93–103

    work page 2023

  24. [24]

    1, 21–30

    , The monogenicity of power-compositional characteristic po lynomials, Albanian Journal of Mathematics 18 (2024), no. 1, 21–30

    work page 2024

  25. [25]

    Jones and D

    L. Jones and D. White, Monogenic trinomials with non-squarefree discriminant , International Journal of Mathematics 32 (2021), 2150089

    work page 2021

  26. [26]

    S. Kaur, S. Kumar, and L. Remete, On the index of power compositional polynomials , Finite Fields and Their Applications 107 (2025), 1071–5797. 14

    work page 2025

  27. [27]

    K. S. Kedlaya, A construction of polynomials with squarefree discriminan ts, Proceedings of the American Mathematical Society 140 (2012), no. 9, 3025–3033. MR 2917075

    work page 2012

  28. [28]

    König, H

    J. König, H. Smith, and Z. W olske, Critical point criteria and dynamically monogenic polynomi als, 2025

    work page 2025

  29. [29]

    Li, On number fields towers defined by iteration of polynomials , Archiv der Mathematik 119 (2022), 371–379

    R. Li, On number fields towers defined by iteration of polynomials , Archiv der Mathematik 119 (2022), 371–379

    work page 2022

  30. [30]

    Pastén, The abc conjecture, arithmetic progressions of primes and sq uarefree values of polynomials at prime arguments , International Journal of Number Theory 11 (2015), no

    H. Pastén, The abc conjecture, arithmetic progressions of primes and sq uarefree values of polynomials at prime arguments , International Journal of Number Theory 11 (2015), no. 3, 721–737. MR 3327840

    work page 2015

  31. [31]

    Sharma and R

    H. Sharma and R. Sarma, Monogenity of composition of polynomials , Ramanujan J. 67 (2025), no. 2, Paper No. 45, 15

    work page 2025

  32. [32]

    Sharma, R

    H. Sharma, R. Sarma, and S. Laishram, Monogenity of iterates of irreducible binomials , Communications in Algebra 52 (2024), 4678–4684

    work page 2024

  33. [33]

    Smith, Radical dynamical monogenicity , Journal de Théorie des Nombres de Bordeaux 37 (2025), 153–169

    H. Smith, Radical dynamical monogenicity , Journal de Théorie des Nombres de Bordeaux 37 (2025), 153–169

    work page 2025

  34. [34]

    Smith and Z

    H. Smith and Z. W olske, Iterates of quadratics and monogenicity , Rocky Mountain Journal of Mathematics (to appear) (2026)

    work page 2026

  35. [35]

    Uchida, When is Z[α ] the ring of the integers? , Osaka Math

    K. Uchida, When is Z[α ] the ring of the integers? , Osaka Math. J. 14 (1977), 155–157. (Jakhar) Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamil Nadu, India- 600036 Email address : Anujjakhar@iitm.ac.in (Kalwaniya) Department of Mathematics, Indian Institute of Technology Madras, Chennai-600036, Tamil Nadu, India Email add...

    work page 1977