Pith. sign in

REVIEW 2 major objections 3 minor 2 cited by

Monogenic trinomials with non-squarefree discriminant

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A formula counts monogenic trinomials with non-squarefree discriminant.

desk verdict The main theorem overclaims at n=2, but the construction for n≥3 is solid and worth a serious referee. read the letter →

arxiv 1908.07947 v2 pith:R753PPAS submitted 2019-08-21 math.NT

Lenny Jones , Daniel White This is my paper · ORCID
classification math.NT MSC 11R0411R0911R3212F12
keywords monogenicpolynomialstrinomialsnon-squarefreediscriminantGaloisgroupindexofanalgebraicintegersquarefreevaluesasymptoticdensityarithmeticprogressions
topics The abc Conjecture
open problems The abc Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that being monogenic does not force the discriminant to be squarefree. For every $n\ge 2$ and every proper divisor $m$ of $n$, it constructs infinite families of trinomials $f(x)=x^n+Ax^m+A$ that generate the full ring of integers of their number field even though their discriminants are not squarefree, and it counts them: the number of admissible $A\le X$ is asymptotic to $X/(\kappa\zeta(2))$ times two explicit local factors, where $\kappa$ is the squarefree kernel of $m$. A companion result counts the case $A\equiv -1\pmod{\kappa^2}$ when $\kappa\mid 210$, and a third theorem produces infinitely many $B$ for trinomials $x^n+Ax^m+B$ with $A\ne B$, conditionally on the abc conjecture for number fields when $t=n/m\ge 5$. Because the Galois group of these trinomials has order at most $\varphi(m)m^t t!$, the families can have Galois groups much smaller than the symmetric group $S_n$.

What carries the argument

The load-bearing identity is the discriminant formula (Theorem 2.1), which expresses $\Delta(f)$ for $f(x)=x^n+Ax^m+B$ in terms of $n,m,A,B$, together with the five-condition criterion (Theorem 2.2) that decides, from the coefficients alone, which prime divisors of $\Delta(f)$ divide the index $[O_K:\mathbb{Z}[\theta]]$. The paper's Lemma 3.1 makes one of the five clauses vacuously true by requiring $\gcd(A,B)$ to be divisible by the squarefree kernel $\kappa$ of $m$, so that monogenicity reduces to the simultaneous squarefreeness of $B$ and the reduced quantity $D=(t^tB^{t-1}+(1-t)^{t-1}A^t)/\gcd(A,B)^{t-1}$. For $A=B$, this collapses to the condition that $A$ and $D=t^t+(1-t)^{t-1}A$ be squarefree. Counting such pairs is then handled by a new asymptotic (Theorem 3.8) for squarefree values of a linear polynomial in an arithmetic progression, built on the classical asymptotic for squarefree integers in arithmetic progressions.

What would settle it

Enumerate, for a fixed pair $(n,m)$ with $m$ a proper divisor of $n$, all $A\le X$ satisfying the conditions of Corollary 3.3, and compute the index $[O_K:\mathbb{Z}[\theta]]$ directly for the first few trinomials it predicts to be monogenic. If any computed index exceeds 1, the characterization behind Theorem 1.1 fails. Alternatively, compare the exact count at a much larger $X$ than the paper's tables against the main term in (1.2); a discrepancy larger than the stated $O(X^{3/4})$ term would falsify the asymptotic.

Watch

Extended reading notes

Core claim

Fix $n\ge 2$ and a proper divisor $m$ of $n$, write $t=n/m$, and let $\kappa$ be the squarefree kernel of $m$. The paper proves that, up to an $O(X^{3/4})$ error, the number of $A\le X$ with $A\equiv 0\pmod{\kappa}$ for which $f(x)=x^n+Ax^m+A$ is monogenic with non-squarefree discriminant is $$\frac{X}{\kappa\zeta(2)}\prod_{p\mid\kappa}\left(1-\frac{1}{p+1}\right)\prod_{p\nmid t(t-1)\kappa}\left(1-\frac{1}{$p^{2}$-1}\right).$$ The characterization behind the count is that, under these hypotheses, monogenicity is equivalent to $A$ and $D=t^t+(1-t)^{t-1}A$ both being squarefree. The same squarefree-pair mechanism, with extra coprimality checks limited to the primes $2,3,5,7$, yields the companion count for $A\equiv -1\pmod{\kappa^2}$. A separate construction with $A\ne B$ gives infinitely many $B$ making $x^n+Ax^m+B$ monogenic, unconditional for $2\le t\le 4$ and conditional on the abc conjecture for number fields for $t\ge 5$; when $m\ge2$, infinitely many of those also have non-squarefree discriminant.

Load-bearing premise

The load-bearing assumption is that the quoted five-condition test fully and correctly decides when a trinomial generates the full ring of integers of its number field; every claimed monogenic family depends on that test being complete. A secondary external assumption, used only for part of one theorem, is the abc conjecture for number fields.

Editorial extensions

If this is right

  • For $m=1$, the condition $A\equiv 0\pmod{\kappa}$ is vacuous, so Theorem 1.1 gives a complete classification of monogenic polynomials of the form $x^n+Ax+A$ with $A\ge 2$, essentially all of which have non-squarefree discriminant.
  • For $n=2^k t$ and $m=2^k$ with $t\ge 2$, the results combine to give a complete classification of monogenic trinomials $x^{2^k t}+A x^{2^k}+A$ with $A\ge 2$, all with non-squarefree discriminant.
  • Every resulting field has Galois group of order at most $\varphi(m)m^t t!$, so taking $n$ large with $m$ commensurate with $n$ makes the Galois group far smaller than $S_n$.
  • For fixed $A$ satisfying the stated divisibility condition, there are infinitely many $B$ such that $x^n+Ax^m+B$ is monogenic; when $m\ge 2$, infinitely many of those also have non-squarefree discriminant, with the $t\ge5$ cases conditional on the abc conjecture for number fields.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension, not pursued in the paper, is to replace the residue conditions $A\equiv0\pmod{\kappa}$ or $A\equiv-1\pmod{\kappa^2}$ by other residue classes modulo powers of $\kappa$; the same squarefree-pair counting method should yield analogous Euler-product constants whenever the corresponding coprimality calculation can be completed.
  • The small-Galois-group bound suggests these trinomials could serve as explicit generators for number fields with prescribed small Galois groups and non-squarefree discriminant, but the paper does not itself construct such fields.
  • When $t(t-1)$ has many small prime factors, the asymptotic constant is close to $1/(\kappa\zeta(2))$, hinting that almost every admissible $A$ yields a monogenic trinomial; checking exact counts at larger $X$ than the tables provide would be a direct numerical test.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper constructs infinite families of monogenic trinomials f(x)=x^n+A x^m+A with non-squarefree discriminant for n≥2 and m a proper divisor of n, and gives asymptotic counts for A≤X. The proofs combine the Jakhar–Khanduja–Sangwan criterion for monogeneity of trinomials, Swan's discriminant formula, and squarefree sieve estimates (Prachar, Helfgott–Hooley–Pasten). A separate result, conditional on the abc conjecture for number fields when n/m≥5, treats the case A≠B. The paper also proves a Galois-group bound showing these families can have Galois groups much smaller than S_n.

Significance. If the main theorems are correct, the paper provides genuinely new infinite families of monogenic trinomials with non-squarefree discriminant and small Galois groups, with explicit densities. The proofs are detailed and the central technique—reducing monogeneity to squarefree conditions on A and an associated linear form D—is transparent and reproducible. The use of the JKS criterion is appropriate, and the analytic number theory inputs are standard. However, the main counting theorems are false as stated for n=2, m=1, because the asymptotic includes odd A for which the discriminant is squarefree. This is a load-bearing error in the paper's principal claims, though it appears to be repairable by restricting to n≥3 or by imposing a parity condition for n=2.

major comments (2)
  1. [Theorem 1.1 and §4.1] Theorem 1.1 is false as stated for n=2, m=1. In that case κ=1, t=2, and Corollary 3.3 gives that f(x)=x^2+Ax+A is monogenic iff A and D=4-A are squarefree, while the discriminant is non-squarefree only if A is even. The proof of Theorem 1.1 applies Theorem 3.8 to count all squarefree A with A-4 squarefree, making no parity restriction. For A=5, A and 4-A are squarefree, so f is monogenic, and Swan's formula gives Δ(f)=5, which is squarefree; yet A=5 is counted by the asymptotic (1.2). Thus the counted set is not contained in the set of trinomials with non-squarefree discriminant. The statement and proof need either the restriction n≥3, or, for n=2,m=1, the additional condition that A is even together with a corresponding adjustment of the asymptotic.
  2. [Theorem 1.2 and §4.2] The same defect appears in Theorem 1.2 when n=2, m=1. Then κ=1, the congruence A≡-1 (mod κ²) is vacuous, and Proposition 3.5 again only guarantees non-squarefree discriminant for even A when t=2 and m=1. The proof counts all squarefree A with A-4 squarefree, including odd values such as A=5 with Δ(f)=5. Consequently, the asymptotic (1.3) overcounts the stated family. The theorem requires the same correction as Theorem 1.1: either restrict to n≥3 or add the appropriate parity condition.
minor comments (3)
  1. [§4.1] In the proof of Theorem 1.1, the application of Theorem 3.8 uses the variable y=a=A/κ and therefore counts a up to X/κ, but the paper does not explicitly say that Theorem 3.8 is applied with X replaced by X/κ; the notation is understandable but would benefit from clarification.
  2. [Theorem 3.8 proof, Eq. (3.10)] The transition from the first line of (3.10) to the second line replaces the upper limit √F(X) with X0 and writes O(X/X0) for the tail; this is correct after summing X/d² over d>X0, but the reader must fill in that step. A sentence explaining the tail bound would improve readability.
  3. [Throughout] The phrase 'non-squarefree discriminant' is occasionally used where 'monogenic with non-squarefree discriminant' is meant; the distinction matters for the n=2 issue, and aligning terminology with the theorem statements would prevent confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; the main derivations are self-contained reductions to external number-theoretic criteria.

full rationale

The paper's asymptotic results (Theorems 1.1 and 1.2) are obtained by applying the paper's own Theorem 3.8, which is proved in the text from Prachar's asymptotic via a standard squarefree-detection argument; the constants in the Euler products are computed from the hypotheses, not fitted to the numerical tables. The monogenicity characterizations (Corollary 3.3, Propositions 3.5 and 3.6) are proved from Swan's discriminant formula and the externally cited JKS criterion (Theorem 2.2), whose assumptions do not include the target result. The authors' previous papers appear only as context or comparison, not as load-bearing support for the main theorems. There is no parameter that is fitted to data and then renamed as a prediction, and no definition is circularly made in terms of the claimed output. A separate correctness concern is visible: Theorem 1.1 as stated at n=2 (and also for A=1 in some m=1 cases) counts squarefree A and D without enforcing the additional conditions in Corollary 3.3 needed for non-squarefree discriminant. That is an error in the implication from Corollary 3.3, not a circularity: the counted set is not by construction equal to the set of claimed polynomials. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims depend on standard external theorems, namely Swan, JKS, Prachar, and squarefree-at-primes asymptotics, plus the paper's own Theorem 3.8. No free parameters are fitted: all asymptotic constants are explicit products over primes derived from Euler products. No new physical or mathematical entities are introduced. The only unproved external input is the abc conjecture for number fields, which is used only for the conditional part of Theorem 1.3 and is clearly flagged.

assumptions (5)
  • standard math Swan's discriminant formula: Delta(x^n + A x^m + B) = (-1)^{n(n-1)/2} B^{m-1} (n^{n/d} B^{(n-m)/d} - (-1)^{n/d} (n-m)^{(n-m)/d} m^{m/d} A^{n/d})^d.
    Invoked repeatedly in Lemma 3.1 and the corollaries to compute Delta(f) and to force the divisibility conditions needed in the JKS criterion. External classical result.
  • standard math JKS characterization: a prime p dividing Delta(f) does not divide [O_K : Z[theta]] iff five coefficient conditions (statements (1)-(5)) hold.
    This is the load-bearing external theorem. All monogenicity conclusions in Sections 3 and 4 are reductions to these five conditions, cited from [20].
  • standard math Prachar's asymptotic theorem on squarefree integers in arithmetic progressions and its corollary for squarefree integers coprime to a fixed q.
    Used as the input for the squarefree sieve in the proof of Theorem 3.8.
  • domain assumption Asymptotic for squarefree values of polynomials at primes, N_F(x) sim c_F x / log x, unconditional when max degree is at most 3 and conditional on the abc conjecture for number fields otherwise.
    Used in the proof of Theorem 1.3 via Corollary 2.7 to produce infinitely many primes p with F(p) squarefree.
  • domain assumption The abc conjecture for number fields, assumed for t >= 5 in Theorem 1.3.
    Explicitly conditional: the theorem statement says the result is unconditional when 2 <= t <= 4 and conditional on the abc conjecture for number fields when t >= 5. The authors flag this rather than presenting it as proven.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Monogenic trinomials with non-squarefree discriminant." pith.science (2026). https://pith.science/paper/R753PPAS

@misc{pith2026190807947,
  author       = {Pith},
  title        = {Pith review of: Monogenic trinomials with non-squarefree discriminant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R753PPAS}},
  note         = {Machine review of arXiv:1908.07947}
}
abstract

For each integer $n\ge 2$, we identify new infinite families of monogenic trinomials $f(x)=x^n+Ax^m+B$ with non-squarefree discriminant, many of which have small Galois group. These families are thus different from many previous examinations of specific trinomial forms in the literature. Moreover, in certain situations when $A=B\ge 2$ with fixed $n$ and $m$, we produce asymptotics on the number of such trinomials with $A\le X$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Discriminants of Fields Generated by Polynomials of Given Height

    math.NT 2019-08 conditional novelty 6.0 of 10

    New upper bounds on how many monic integer polynomials of degree n and height H share a fixed field discriminant, plus improved lower bounds for distinct discriminants from trinomials.

  2. Monogenic fields arising from trinomials

    math.NT 2019-08 conditional novelty 5.0 of 10

    For two families of trinomials, this paper proves explicit congruence criteria for monogeneity in degrees 5 and 6 and shows infinite monogenic subfamilies for every n greater than 2 with positive density.

Reference graph

Works this paper leans on

35 extracted references · 33 canonical work pages · cited by 2 Pith papers

  1. [1]

    Ahmad, T

    S. Ahmad, T. Nakahara and A. Hameed, On certain pure sextic fields related to a problem of Hasse , Internat. J. Algebra Comput. 26 (2016), no. 3, 577–583

    work page 2016

  2. [2]

    Ahmad, T

    S. Ahmad, T. Nakahara and S. M. Husnine, Power integral bases for certain pure sextic fields , Int. J. Number Theory 10 (2014), no. 8, 2257–2265

    work page 2014

  3. [3]

    Alaca and K

    S ¸. Alaca and K. S. Williams, Introductory algebraic number theory, Cambridge University Press , Cambridge, 2004

    work page 2004

  4. [4]

    Bhargava, A

    M. Bhargava, A. Shankar and X. Wang, Squarefree values of polynomial dis- criminants I , https://arxiv.org/abs/1611.09806. 18 LENNY JONES AND DANIEL WHITE

    arXiv

  5. [5]

    D. W. Boyd, G. Martin and M. Thom, Squarefree values of trinomial discrim- inants, LMS J. Comput. Math. 18 (2015), no. 1, 148–169

    work page 2015

  6. [6]

    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

  7. [7]

    D. Eloff, B. Spearman and K. Williams, A4-sextic fields with a power basis , Missouri Journal of Mathematical Sciences 19 (2007), 188–194

    work page 2007

  8. [8]

    Erd˝ os,Arithmetical properties of polynomials , The Journal of the London Mathematical Society, 28 (1953), 416—425

    P. Erd˝ os,Arithmetical properties of polynomials , The Journal of the London Mathematical Society, 28 (1953), 416—425

    work page 1953

Show all 35 references
  1. [9]

    Estermann, Einige S¨ atze ¨ uber quadratfreie Zahlen, Math

    T. Estermann, Einige S¨ atze ¨ uber quadratfreie Zahlen, Math. Ann., 105 (1931), 653–662

    1931

  2. [10]

    Ga´ al, Diophantine equations and power integral bases

    I. Ga´ al, Diophantine equations and power integral bases. New comput ational methods., Birkh¨ auser Boston, Inc., Boston, MA, (2002)

    2002

  3. [11]

    Ga´ al and L

    I. Ga´ al and L. Remete, Power integral bases in a family of sextic fields with quadratic subfields , Tatra Mt. Math. Publ. 64 (2015), 59–66

    2015

  4. [12]

    Ga´ al and L

    I. Ga´ al and L. Remete,Integral bases and monogenity of pure fields , J. Number Theory 173 (2017), 129–146

    2017

  5. [13]

    T. A. Gassert, Discriminants of Chebyshev radical extensions , J. Th´ eor. Nom- bres Bordeaux 26 (2014), no. 3, 607–634

    2014

  6. [14]

    Guersenzvaig, Elementary criteria for irreducibility of f (xr), Israel J

    N. Guersenzvaig, Elementary criteria for irreducibility of f (xr), Israel J. Math. 169 (2009), 109–123

    2009

  7. [15]

    H. A. Helfgott, On the square-free sieve , Acta Arith. 115 (2004), 349–402

    2004

  8. [16]

    H. A. Helfgott, Square-free values of f (p), f cubic., Acta Math. 213 (2014), no. 1, 107–135

    2014

  9. [17]

    Hooley, On the power free values of polynomials , Mathematika 14 (1967), 21–26

    C. Hooley, On the power free values of polynomials , Mathematika 14 (1967), 21–26

    1967

  10. [18]

    Hooley, Applications of Sieve Methods to the Theory of Numbers , Cambridge Tracts in Mathematics, No

    C. Hooley, Applications of Sieve Methods to the Theory of Numbers , Cambridge Tracts in Mathematics, No. 70. Cambridge University Pre ss, Cambridge-New York-Melbourne, (1976). xiv+122 pp

    1976

  11. [19]

    Huard, B

    J. Huard, B. Spearman and K. Williams, Integral bases for quartic fields with quadratic subfields , J. Number Theory 51 (1995), no. 1, 87–102

    1995

  12. [20]

    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

  13. [21]

    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. MONOGENIC TRINOMIALS WITH NON-SQUAREFREE DISCRIMINANT 19

    2017

  14. [22]

    Jhorar and S

    B. Jhorar and S. Khanduja, On power basis of a class of algebraic number fields, Int. J. Number Theory 12 (2016), no. 8, 2317–2321

    2016

  15. [23]

    Jones, Monogenic polynomials with non-squarefree discriminant , Proc

    L. Jones, Monogenic polynomials with non-squarefree discriminant , Proc. Amer. Math. Soc. 148 (2020), no. 4, 1527–1533

    2020

  16. [24]

    Jones, Some new infinite families of monogenic polynomials with non - squarefree discriminant, Acta Arith

    L. Jones, Some new infinite families of monogenic polynomials with non - squarefree discriminant, Acta Arith. (to appear)

  17. [25]

    Jones and T

    L. Jones and T. Phillips, Infinite families of monogenic trinomials and their Galois groups , Internat. J. Math. 29 (2018), no. 5, 1850039, 11 pp

    2018

  18. [26]

    Kable, Power bases in dihedral quartic fields , J

    A. Kable, Power bases in dihedral quartic fields , J. Number Theory 76 (1999), no. 1, 120–129

    1999

  19. [27]

    Kedlaya, A construction of polynomials with squarefree discriminan ts, Proc

    K. Kedlaya, A construction of polynomials with squarefree discriminan ts, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3025–3033

    2012

  20. [28]

    Nagel, Zur Arithmetik der Polynome , Abh

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

    1922

  21. [29]

    Narkiewicz, Elementary and analytic theory of algebraic numbers, Third edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, (2 004)

    W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Third edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, (2 004)

  22. [30]

    Pasten, The ABC conjecture, arithmetic progressions of primes and s quare- free values of polynomials at prime arguments , Int

    H. Pasten, The ABC conjecture, arithmetic progressions of primes and s quare- free values of polynomials at prime arguments , Int. J. Number Theory 11 (2015), no. 3, 721–737

    2015

  23. [31]

    Prachar, ¨Uber die kleinste quadratfreie Zahl einer arithmetischen R eihe, Monatsh

    K. Prachar, ¨Uber die kleinste quadratfreie Zahl einer arithmetischen R eihe, Monatsh. Math. 62 (1958), 173–176

    1958

  24. [32]

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

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

    2000

  25. [33]

    I. E. Shparlinski, Squarefree parts of discriminants of trinomials , Arch. Math. (Basel) 102 (2014), no. 6, 545–554

    2014

  26. [34]

    B. K. Spearman, Monogenic A4 quartic fields , International Mathematical Forum, 1, (2006), no. 40, 1969–1974

    2006

  27. [35]

    Swan Factorization of polynomials over finite fields , Pacific J

    R. Swan Factorization of polynomials over finite fields , Pacific J. Math. 12 (1962), 1099–1106. Shippensburg University, Department of Mathematics, 1871 O ld Main Drive, Ship- pensburg, PA 17257 Email address : lkjone@ship.edu Bryn Mawr College, Department of Mathematics, 101 No...

    1962

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.