pith. sign in

arxiv: 2604.18830 · v1 · submitted 2026-04-20 · 🧮 math.NT

Characterizing monogenic trinomials boldsymbol{x¹²+ax⁶+b} according to their Galois groups

Lenny Jones This is my paper

Pith reviewed 2026-05-10 03:12 UTC · model grok-4.3

classification 🧮 math.NT
keywords monogenic trinomialsGalois groupsnumber fieldsintegral basesirreducible polynomialsalgebraic integerstrinomial polynomialsGalois theory
0
0 comments X

The pith

All monogenic trinomials of the form x^{12} + a x^6 + b are described explicitly by their Galois groups over Q.

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

The paper seeks to classify every irreducible monogenic trinomial f(x) = x^{12} + a x^6 + b with nonzero integer coefficients a and b. Monogenic means that a root theta generates a power integral basis for the ring of integers of Q(theta). Classification proceeds by partitioning the polynomials according to the Galois group G of f over Q and supplying explicit lists or parametric forms for the monogenic members of each group. A reader cares because these descriptions make concrete the arithmetic of the associated degree-12 number fields, including their integral bases and splitting behavior, and extend similar complete lists already known for degree-4 and degree-6 trinomials of the same shape.

Core claim

For each possible Galois group G of f(x) over Q, explicit descriptions are given of all monogenic trinomials f(x) having Galois group G.

What carries the argument

The Galois group G of the trinomial over Q, which is used to organize and enumerate the monogenic examples in explicit families.

If this is right

  • Every monogenic example in the family is captured in one of the explicit lists or parametric forms attached to its Galois group.
  • The classification is exhaustive for this degree-12 power-compositional trinomial shape.
  • The same style of description now exists for the quartic, sextic, and dodecic cases of the same trinomial shape.

Where Pith is reading between the lines

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

  • The lists supply concrete generators for number fields whose Galois groups and integral bases are both prescribed in advance.
  • The same partitioning technique may apply directly to other composite exponents or to trinomials with more terms.
  • Once the lists are known, one can test statistical questions such as the density of monogenic fields inside each Galois group for this family.

Load-bearing premise

The possible Galois groups that polynomials of this exact form can realize are already known completely, and explicit criteria exist that detect irreducibility and the monogenic property without omissions.

What would settle it

An explicit monogenic trinomial x^{12} + a x^6 + b (with ab nonzero) whose Galois group over Q is not among the groups treated by the descriptions, or a polynomial that belongs to one of the groups yet fails to appear in the listed families.

read the original abstract

Let $f(x)=x^{12}+ax^{6}+b\in {\mathbb Z}[x]$, with $ab\ne 0$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{11}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. These results extend recent work on monogenic power-compositional quartic and sextic trinomials.

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 / 3 minor

Summary. The manuscript studies irreducible monogenic trinomials f(x) = x^{12} + a x^6 + b in Z[x] with ab ≠ 0. It determines the attainable Galois groups G of f over Q and, for each such G, supplies explicit arithmetic conditions on the coefficients a and b that characterize precisely the monogenic trinomials realizing that group. The work extends analogous classifications previously obtained for power-compositional quartic and sextic trinomials.

Significance. If the explicit descriptions hold, the paper delivers a complete classification of monogenic number fields generated by this trinomial family according to their Galois groups. Such results are valuable in computational algebraic number theory for constructing fields with prescribed Galois groups and power integral bases; the sparse trinomial shape permits concrete use of resolvents and discriminant formulas that reduce the monogenic condition to coefficient constraints.

minor comments (3)
  1. [Abstract] Abstract: the statement that explicit descriptions are given for each possible Galois group would be strengthened by indicating the number of attainable groups or including a brief enumeration.
  2. [Sections containing the main theorems] The proofs of the coefficient conditions would benefit from one or two fully worked numerical examples that verify both monogenicity and the Galois group simultaneously.
  3. [Introduction and §3] Notation for the quadratic resolvent and its factorization types could be made uniform across the introduction and the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its significance for computational algebraic number theory, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No circularity: explicit coefficient conditions derived from standard Galois resolvents and monogenicity criteria

full rationale

The paper determines attainable Galois groups for the family via the quadratic resolvent y² + a y + b and then enumerates arithmetic conditions on a, b that simultaneously enforce irreducibility, monogenicity (via discriminant-index checks), and Gal(f) = G. These conditions are obtained by direct computation on the sparse trinomial shape rather than by fitting parameters or renaming prior outputs. Reliance on earlier quartic/sextic results is limited to methodological analogy and does not supply the degree-12 lists; the central output remains an independent enumeration of coefficient constraints. No self-definitional loop, fitted-input prediction, or load-bearing self-citation chain appears in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The classification rests on the standard axioms of Galois theory and algebraic number theory together with the assumption that the possible transitive subgroups of S_12 that can arise for this trinomial shape are already enumerated in the literature.

axioms (2)
  • domain assumption The Galois group of an irreducible trinomial of this shape is a transitive subgroup of S_12 that preserves the power-compositional structure.
    Invoked when the paper restricts attention to possible G that can occur for f(x) = x^{12} + a x^6 + b.
  • standard math Standard criteria for monogenicity (index of the power basis equals 1) can be applied via discriminant or resultant computations.
    Used to translate the monogenic condition into explicit arithmetic conditions on a and b.

pith-pipeline@v0.9.0 · 5430 in / 1366 out tokens · 27990 ms · 2026-05-10T03:12:16.730345+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

31 extracted references · 31 canonical work pages

  1. [1]

    Awtrey and P

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

    work page 2020

  2. [2]

    Awtrey and A

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

    work page 2024

  3. [3]

    Bremner and B

    A. Bremner and B. Spearman,Cyclic sextic trinomialsx 6 +Ax+B, Int. J. Number Theory 6(2010), no. 1, 161–167

    work page 2010

  4. [4]

    Brown, B

    S. Brown, B. Spearman and Q. Yang,On sextic trinomials with Galois groupC 6,S 3 or C3 ×S 3, J. Algebra Appl.12(2013), no. 1, 1250128, 9 pp

    work page 2013

  5. [5]

    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

    work page 2010

  6. [6]

    Butler and J

    G. Butler and J. McKay,The transitive groups of degree up to eleven, Comm. Algebra11 (1983), no. 8, 863–911

    work page 1983

  7. [7]

    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

  8. [8]

    Chen,Elementary characterization for Galois groups ofx 12 +ax 6 +b,arXiv:2410

    M. Chen,Elementary characterization for Galois groups ofx 12 +ax 6 +b,arXiv:2410. 00870v2

    work page

  9. [9]

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

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

    work page 2000

  10. [10]

    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

    work page 2017

  11. [11]

    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

    work page 2025

  12. [12]

    Harrington and L

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

    work page 2025

  13. [13]

    Harrington and L

    J. Harrington and L. Jones,Monogenic trinomials of the formx 4 +ax 3 +dand their Galois groups, J. Algebra Appl. (to appear). MONOGENIC TRINOMIALSx 12 +ax 6 +b15

    work page

  14. [14]

    Harrington and L

    J. Harrington and L. Jones,Monogenic sextic trinomialsx 6 +Ax 3 +Band their Galois groups, Acta Arith. (to appear)

    work page

  15. [15]

    Harrington and L

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

    work page

  16. [16]

    Jakhar, S

    A. Jakhar, S. Khanduja and N. Sangwan,Characterization of primes dividing the index of a trinomial, Int. J. Number Theory13(2017), no. 10, 2505–2514

    work page 2017

  17. [17]

    Jones,Sextic reciprocal monogenic dihedral polynomials, Ramanujan J.56(2021), no

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

    work page 2021

  18. [18]

    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

    work page 2021

  19. [19]

    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

    work page 2022

  20. [21]

    Jones,Monogenic cyclic trinomials of the formx 4 +cx+d, Acta Arith.218(2025), no

    L. Jones,Monogenic cyclic trinomials of the formx 4 +cx+d, Acta Arith.218(2025), no. 4, 385–394

    work page 2025

  21. [22]

    Jones,Monogenic even cyclic sextic polynomials, Math

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

    work page 2025

  22. [23]

    Jones,Monogenic cyclic cubic trinomials, Bol

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

    work page 2025

  23. [24]

    Jones,Monogenic even quartic trinomials, Bull

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

    work page 2025

  24. [25]

    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

    work page 2025

  25. [26]

    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

    work page 2021

  26. [27]

    Kappe and B

    L-C. Kappe and B. Warren,An elementary test for the Galois group of a quartic polynomial, Amer. Math. Monthly96(1989), no. 2, 133–137

    work page 1989

  27. [28]

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

    work page 2025

  28. [29]

    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 Kˆ okyˆ uroku Bessatsu, B12, Res. Inst. Math. Sci. (RIMS), Kyoto, (2009)

    work page 2007

  29. [30]

    Swan,Factorization of polynomials over finite fields, Pacific J

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

    work page 1962

  30. [31]

    Voutier,Families of cyclic quartic monogenic polynomials, Acta Arith.219, No

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

    work page 2025

  31. [32]

    L. C. Washington,Introduction to cyclotomic fields, Second edition, Graduate Texts in Math- ematics,83, Springer-Verlag, New York, 1997. Professor Emeritus, Department of Mathematics, Shippensburg University, Shippens- burg, Pennsylvania 17257, USA Email address, Lenny Jones:doctorlennyjones@gmail.com

    work page 1997