pith. sign in

arxiv: 2605.25753 · v1 · pith:WS6KF4PJnew · submitted 2026-05-25 · 🧮 math.NT

A Note on Abelian Monogenic Trinomials

Lenny Jones This is my paper

Pith reviewed 2026-06-29 20:45 UTC · model grok-4.3

classification 🧮 math.NT
keywords abelian Galois groupmonogenic polynomialtrinomialring of integersnumber fieldindex computationGalois group of trinomial
0
0 comments X

The pith

All abelian monogenic trinomials of the form x^{2n} + a x^n + b are completely determined.

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

The paper sets out to classify every monic irreducible trinomial f(x) = x^{2n} + a x^n + b over the integers such that its Galois group over the rationals is abelian and the powers of a root form a basis for the ring of integers of the number field it generates. A reader would care because these conditions produce number fields with restricted symmetries and a simple integral basis that simplifies arithmetic computations. The work proceeds by partitioning the possible abelian Galois groups that can arise for trinomials of this shape and then checking the monogenic condition through explicit index calculations in each case.

Core claim

An abelian monogenic trinomial is a monic irreducible polynomial f(x) = x^{2n} + a x^n + b in Z[x] with n at least 1 and ab not zero, whose splitting field over Q has abelian Galois group and for which the ring of integers of Q(θ) equals Z[θ] where f(θ) = 0. The paper determines every such trinomial by exhaustive analysis of the possible abelian Galois groups together with direct verification of the index being one.

What carries the argument

Case-by-case examination of abelian Galois groups that can occur for trinomials x^{2n} + a x^n + b combined with index computations that test whether the power basis generates the full ring of integers.

If this is right

  • The possible values of n, a, and b are restricted to those for which the Galois group is abelian and the index equals one.
  • Each such trinomial produces an explicit abelian extension of Q of degree 2n that admits a power integral basis.
  • For any fixed n the search reduces to checking only finitely many candidate pairs a, b.
  • These polynomials supply concrete generators for all abelian monogenic fields that arise from trinomials of this biquadratic shape.

Where Pith is reading between the lines

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

  • The resulting list can be used to produce all examples of such fields for small values of n by direct substitution of the allowed a and b.
  • Similar exhaustive methods could be applied to classify monogenic trinomials whose Galois groups are solvable but not necessarily abelian.
  • The classification supplies test cases for conjectures about the density of monogenic fields inside families with fixed Galois group.

Load-bearing premise

That every possible abelian Galois group for these trinomials can be enumerated and that the index calculations in each case are sufficient to detect all monogenic examples without omissions.

What would settle it

An explicit integer triple n, a, b with n at least 1 and ab nonzero such that x^{2n} + a x^n + b is irreducible over Q, has abelian Galois group, generates a monogenic ring of integers, yet fails to appear in the paper's final list.

read the original abstract

An abelian monogenic polynomial $f(x)\in {\mathbb Z}[x]$ is a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$, such that the Galois group of $f(x)$ over ${\mathbb Q}$ is abelian, and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we determine all abelian monogenic trinomials of the form $x^{2n}+ax^{n}+b$, where $n,a,b\in {\mathbb Z}$ with $n\ge 1$ and $ab\ne 0$.

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

0 major / 2 minor

Summary. The manuscript determines all abelian monogenic trinomials of the form x^{2n} + a x^n + b (n ≥ 1, ab ≠ 0) that are irreducible over Q, have abelian Galois group over Q, and for which {1, θ, …, θ^{2n-1}} forms a Z-basis of the ring of integers of Q(θ). The classification proceeds by restricting possible Galois groups of the biquadratic or higher even-degree extensions and computing the index of the order Z[θ] in the full ring of integers.

Significance. A complete explicit classification of this family supplies a finite list of concrete polynomials that can be used to test conjectures on monogenic fields with abelian Galois groups and to generate examples for computational verification of integral bases. The result is of moderate interest within algebraic number theory but does not introduce new methods beyond standard Galois-theoretic and index computations.

minor comments (2)
  1. The abstract and introduction should explicitly state the degree 2n and clarify that the Galois group is taken over Q; the current wording leaves the precise Galois closure implicit.
  2. Theorem statements would benefit from a short table or enumerated list of the surviving (n,a,b) triples rather than a purely descriptive paragraph, to facilitate verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The summary provided accurately captures the scope and methods of our classification of abelian monogenic trinomials of the indicated form.

Circularity Check

0 steps flagged

No circularity; standard classification via Galois and index analysis

full rationale

The paper defines abelian monogenic trinomials and states it determines all such trinomials of the given form. This is a classification result relying on Galois group restrictions and index computations for the ring of integers. No equations reduce a claimed prediction to a fitted input by construction, no self-citations bear the central load as uniqueness theorems, and no ansatz or renaming is smuggled in. The argument structure is the standard one for such results and remains independent of the output list of polynomials.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard definitions from algebraic number theory; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption The polynomial is monic and irreducible over the rationals
    Explicitly required by the definition of an abelian monogenic polynomial given in the abstract.
  • domain assumption The Galois group over Q is abelian
    Part of the definition of abelian monogenic polynomial stated in the abstract.

pith-pipeline@v0.9.1-grok · 5638 in / 1190 out tokens · 46834 ms · 2026-06-29T20:45:11.380764+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

68 extracted references · 3 canonical work pages · 1 internal anchor

  1. [1]

    Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag , 2000

    H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag , 2000

    2000

  2. [2]

    U ber die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlk\

    G. Degert, \" U ber die Bestimmung der Grundeinheit gewisser reell-quadratischer Zahlk\" o rper , Abh. math. Sem. Univ. Hamburg 22 (1958), 92--97

    1958

  3. [3]

    L. E. Dickson, The Galois Group of a Reciprocal Quartic Equation,

  4. [4]

    Erd o s, Arithmetical properties of polynomials , J

    P. Erd o s, Arithmetical properties of polynomials , J. London Math. Soc. 28 (1953), 416--425

    1953

  5. [5]

    Fl\' o rez, R

    R. Fl\' o rez, R. Higuita and A. Ram\' i rez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials , J. Integer Seq. 22 (2019), no. 4, Art. 19.4.4, 28 pp

    2019

  6. [6]

    Gajdzica, Discriminants of special quadrinomials ,

    K. Gajdzica, Discriminants of special quadrinomials ,

  7. [7]

    M.-N. Gras, Non monog\' e n\' e it\' e de l'anneau des entiers de certaines extensions ab\' e lienne de (French) [Nonmonogeneity of the ring of integers of certain abelian extensions of ] Number theory (Besan c on), 1983--1984, Exp. No. 5, 25 pp., Publ. Math. Fac. Sci. Besan c on, Univ. Franche-Comt\' e , Besan c on, 1984

    1983

  8. [8]

    M.-N. Gras, Non monog\' e n\' e it\' e de l'anneau des entiers des extensions cycliques de de degr\' e premier 5 , (French) [Nonmonogeneity of the ring of integers of cyclic extensions of of prime degree 5 ] J. Number Theory 23 (1986), no. 3, 347--353

    1986

  9. [9]

    M.-N. Gras, Condition n\' e cessaire de monog\' e n\' e it\' e de l'anneau des entiers d'une extension ab\' e lienne de (French) [A necessary condition for monogeneity of the ring of integers of an abelian extension of ], S\' e minaire de th\' e orie des nombres, Paris 1984--85, 97--107, Progr. Math., 63 , Birkh\" a user Boston, Boston, MA, 1986

    1984

  10. [10]

    Jones, Monogenic even octic polynomials and their Galois groups, New York J

    L. Jones, Monogenic even octic polynomials and their Galois groups, New York J. Math. 31 (2025), 91--125

    2025

  11. [11]

    Jones, Monogenic reciprocal quartic polynomials and their Galois groups , Vietnam Journal of Mathemaics (to appear)

    L. Jones, Monogenic reciprocal quartic polynomials and their Galois groups , Vietnam Journal of Mathemaics (to appear)

  12. [12]

    Characterizing monogenic trinomials $\boldsymbol{x^{12}+ax^6+b}$ according to their Galois groups

    L. Jones, Characterizing monogenic trinomials x^ 12 +ax^6+b according to their Galois groups , arXiv:2604.18830

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Jones and D

    L. Jones and D. White, Monogenic trinomials with non-squarefree discriminant , Internat. J. Math. 32 (2021), no. 13, Paper No. 2150089, 21 pp

    2021

  14. [14]

    Kappe and B

    L-C. Kappe and B. Warren, An elementary test for the Galois group of a quartic polynomial, Amer. Math. Monthly 96 (1989), no. 2, 133--137

    1989

  15. [15]

    S. Kaur, S. Kumar and L. Remete, On the index of power compositional polynomials , Finite Fields Appl. 107 (2025), Paper No. 102642, 20 pp

    2025

  16. [16]

    Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, (2001)

    T. Koshy, Fibonacci and Lucas numbers with applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, (2001)

    2001

  17. [17]

    Lavallee, B

    M. Lavallee, B. Spearman, and K. Williams, Lifting monogenic cubic fields to monogenic sextic fields, Kodai Math. J. 34 (2011), no. 3, 410--425

    2011

  18. [18]

    Motoda, T

    Y. Motoda, T. Nakahara, A.S.I. Shah and T. Uehara, On a problem of Hasse, Algebraic number theory and related topics 2007, 209--221, RIMS K\^ o ky\^ u roku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, (2009)

    2007

  19. [19]

    Nagel, Zur Arithmetik der Polynome, Abh

    T. Nagel, Zur Arithmetik der Polynome, Abh. Math. Sem. Hamburg. Univ. 1 (1922), 179--194

    1922

  20. [20]

    Prachar, \" U ber die kleinste quadratfreie Zahl einer arithmetischen Reihe (German), Monatsh

    K. Prachar, \" U ber die kleinste quadratfreie Zahl einer arithmetischen Reihe (German), Monatsh. Math. 62 (1958), 173--176

    1958

  21. [21]

    Richaud, Sur la resolution des equations x^2-Ay^2= 1 , Atti Accad

    C. Richaud, Sur la resolution des equations x^2-Ay^2= 1 , Atti Accad. pontif. Nuovi Lincei (1866), 177--182

  22. [22]

    Robbins, Vieta's triangular array and a related family of polynomials , Internat

    N. Robbins, Vieta's triangular array and a related family of polynomials , Internat. J. Math. Math. Sci. 14 (1991), no. 2, 239--244

    1991

  23. [23]

    Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, 77 , Cambridge University Press, Cambridge, 2000

    A. Schinzel, Polynomials with Special Regard to Reducibility, Encyclopedia of Mathematics and its Applications, 77 , Cambridge University Press, Cambridge, 2000

    2000

  24. [24]

    Swan, Factorization of polynomials over finite fields, Pacific J

    R. Swan, Factorization of polynomials over finite fields, Pacific J. Math. 12 (1962), 1099--1106

    1962

  25. [25]

    Voutier, Families of cyclic quartic monogenic polynomials , Acta Arith

    P. Voutier, Families of cyclic quartic monogenic polynomials , Acta Arith. 219 , No. 4, 365--377 (2025)

    2025

  26. [26]

    L. C. Washington, Introduction to cyclotomic fields, Second edition, Graduate Texts in Mathematics , 83 , Springer-Verlag, New York, 1997

    1997

  27. [27]

    Wikipedia, Powerful Number , 9 November 2025, https://en.wikipedia.org/wiki/Powerful_number

    2025

  28. [28]

    Yokoi, On real quadratic fields containing units with norm -1 , Nagoya Math

    H. Yokoi, On real quadratic fields containing units with norm -1 , Nagoya Math. J. 33 (1968), 139--152

    1968

  29. [29]

    Zsigmondy, Zur Theorie der Potenzreste , Journal Monatshefte f\" u r Mathematik, 3 (1): (1892), 265--284

    K. Zsigmondy, Zur Theorie der Potenzreste , Journal Monatshefte f\" u r Mathematik, 3 (1): (1892), 265--284

  30. [30]

    Awtrey and A

    C. Awtrey and A. Lee, Galois groups of reciprocal sextic polynomials , Bull. Aust. Math. Soc. 109 (2024), 37--44

    2024

  31. [31]

    Awtrey and F

    C. Awtrey and F. Patane, On the Galois group of a reciprocal even octic polynomial, Commun. Alg. 52 (2024), no. 7, 3018--3026

    2024

  32. [32]

    Berg\' e , J

    A.-M. Berg\' e , J. Martinet and M. Olivier, The computation of sextic fields with a quadratic subfield, Math. Comp. 54 (1990), no. 190, 869--884

    1990

  33. [33]

    Y. Bilu, G. Hanrot and P. M. Voutier, Existence of primitive divisors of Lucas and Lehmer numbers. With an appendix by M. Mignotte. , J. Reine Angew. Math. 539 (2001), 75--122

    2001

  34. [34]

    Booker and T

    A. Booker and T. D. Browning, Square-free values of reducible polynomials, Discrete Anal. 2016, Paper No. 8, 16 pp

    2016

  35. [35]

    Bremner and B

    A. Bremner and B. Spearman, Cyclic sextic trinomials x^6+Ax+B , Int. J. Number Theory 6 (2010), no. 1, 161--167

    2010

  36. [36]

    Brown, On the Galois groups of sextic trinomials, https://circle.ubc.ca/handle/2429/36998

    S. Brown, On the Galois groups of sextic trinomials, https://circle.ubc.ca/handle/2429/36998

  37. [37]

    Brown, B

    S. Brown, B. Spearman and Q. Yang, On sextic trinomials with Galois group C_6 , S_3 or C_3 S_3 , J. Algebra Appl. 12 (2013), no. 1, 1250128, 9 pp

    2013

  38. [38]

    Brown, B

    S. Brown, B. Spearman and Q. Yang, On the Galois groups of sextic trinomials , JP J. Algebra Number Theory Appl. 18 (2010), no. 1, 67--77

    2010

  39. [39]

    Butler and J

    G. Butler and J. McKay, The transitive groups of degree up to eleven, Comm. Algebra 11 (1983), no. 8, 863--911

    1983

  40. [40]

    Cavallo, An elementary computation of the Galois groups of symmetric sextic trinomials , arXiv:1902.00965v2

    A. Cavallo, An elementary computation of the Galois groups of symmetric sextic trinomials , arXiv:1902.00965v2

    work page arXiv 1902

  41. [41]

    Chen, Elementary characterization for Galois groups of x^ 12 +ax^6+b , arXiv:2410.00870v2

    M. Chen, Elementary characterization for Galois groups of x^ 12 +ax^6+b , arXiv:2410.00870v2

    work page arXiv

  42. [42]

    Awtrey and P

    C. Awtrey and P. Jakes, Galois groups of even sextic polynomials, Canad. Math. Bull. 63 (2020), no. 3, 670--676

    2020

  43. [43]

    Harrington and L

    J. Harrington and L. Jones, The irreducibility of power compositional sextic polynomials and their Galois groups, Math. Scand. 120 (2017), no. 2, 181--194

    2017

  44. [44]

    Harrington and L

    J. Harrington and L. Jones, Some new polynomial discriminant formulas , Lith. Math. J. 61 (2021), no. 4, 483--490

    2021

  45. [45]

    Harrington and L

    J. Harrington and L. Jones, Monogenic cyclotomic compositions , Kodai Math. J. 44 (2021), no. 1, 115--125

    2021

  46. [46]

    Harrington and L

    J. Harrington and L. Jones, Monogenic quartic polynomials and their Galois groups , Bull. Aust. Math. Soc. 111 (2025), no. 2, 244--259

    2025

  47. [47]

    Harrington and L

    J. Harrington and L. Jones, The irreducibility and monogenicity of power-compositional trinomials, Math. J. Okayama Univ. 67 (2025), 53--65

    2025

  48. [48]

    Harrington and L

    J. Harrington and L. Jones, Monogenic trinomials of the form x^4+ax^3+d and their Galois groups , J. Algebra Appl. (to appear)

  49. [49]

    Harrington and L

    J. Harrington and L. Jones, Monogenic sextic trinomials x^6+Ax^3+B and their Galois groups , Acta Arith. (to appear)

  50. [50]

    Harrington and L

    J. Harrington and L. Jones, On the monogenicity and Galois groups of x^ 2p +ax^p+b^p , Bull. Aust. Math. Soc. (to appear)

  51. [51]

    A. F. Horadam, Vieta polynomials, A special tribute to Calvin T. Long, Fibonacci Quart. 40 (2002), no. 3, 223--232

    2002

  52. [52]

    Ide and L

    J. Ide and L. Jones Infinite families of A_4 -sextic polynomials , Canad. Math. Bull. 57 (2014), no. 3, 538--545

    2014

  53. [53]

    Jakhar, R

    A. Jakhar, R. Kalwaniya and S. Kotyada, Explicit discriminant, monogenity and Galois group of a class of polynomials , (2024) hal-04492809

    2024

  54. [54]

    Jakhar, S

    A. Jakhar, S. Khanduja and N. Sangwan, On prime divisors of the index of an algebraic integer , J. Number Theory 166 (2016), 47--61

    2016

  55. [55]

    Jakhar, S

    A. Jakhar, S. Khanduja and N. Sangwan, Characterization of primes dividing the index of a trinomial, Int. J. Number Theory 13 (2017), no. 10, 2505--2514

    2017

  56. [56]

    Jakhar, S

    A. Jakhar, S. Laishram and P. Yadav, Explicit discriminant of a class of polynomial, monogenity and Galois group , (2024) hal-04496946

    2024

  57. [57]

    Janson, Resultant and discriminant of polynomials , https://www2.math.uu.se/ svantejs/papers/sjN5.pdf

    S. Janson, Resultant and discriminant of polynomials , https://www2.math.uu.se/ svantejs/papers/sjN5.pdf

  58. [58]

    Jones, Sextic reciprocal monogenic dihedral polynomials , Ramanujan J

    L. Jones, Sextic reciprocal monogenic dihedral polynomials , Ramanujan J. 56 (2021), no. 3, 1099--1110

    2021

  59. [59]

    Jones, Infinite families of reciprocal monogenic polynomials and their Galois groups , New York J

    L. Jones, Infinite families of reciprocal monogenic polynomials and their Galois groups , New York J. Math. 27 (2021), 1465--1493

    2021

  60. [60]

    Jones, Reciprocal monogenic quintinomials of degree 2^n , Bull

    L. Jones, Reciprocal monogenic quintinomials of degree 2^n , Bull. Aust. Math. Soc. 106 (2022), no. 3, 437--447

    2022

  61. [61]

    Jones, Monogenic reciprocal trinomials and their Galois groups , J

    L. Jones, Monogenic reciprocal trinomials and their Galois groups , J. Algebra Appl. 21 (2022), no. 2, Paper No. 2250026, 11 pp

    2022

  62. [62]

    Jones, On the monogenicity of power-compositional Shanks polynomials , Funct

    L. Jones, On the monogenicity of power-compositional Shanks polynomials , Funct. Approx. Comment. Math. 69 (2023), no. 1, 93--103

    2023

  63. [63]

    Jones, Monogenic even quartic trinomials , Bull

    L. Jones, Monogenic even quartic trinomials , Bull. Aust. Math. Soc. 111 (2025), no. 2, 238--243

    2025

  64. [64]

    Jones, The monogenicity and Galois groups of certain reciprocal quintinomials , Funct

    L. Jones, The monogenicity and Galois groups of certain reciprocal quintinomials , Funct. Approx. Comment. Math. (to appear)

  65. [65]

    Jones, Monogenic cyclic trinomials of the form x^4+cx+d , Acta Arith

    L. Jones, Monogenic cyclic trinomials of the form x^4+cx+d , Acta Arith. 218 (2025), no. 4, 385--394

    2025

  66. [66]

    Jones, Monogenic even cyclic sextic polynomials , Math

    L. Jones, Monogenic even cyclic sextic polynomials , Math. Slovaca, 75 , No. 5, (2025), 1021--1034

    2025

  67. [67]

    Jones, Monogenic cyclic cubic trinomials, Bol

    L. Jones, Monogenic cyclic cubic trinomials, Bol. Soc. Mat. Mex. (3) 31 (2025), no. 1, Paper No. 28

    2025

  68. [68]

    Jones, Monogenic even quartic trinomials, Bull

    L. Jones, Monogenic even quartic trinomials, Bull. Aust. Math. Soc. 111 (2025), no. 2, 238--243

    2025