Irreducibility criterion for certain trinomials

A.Satyanarayana Reddy; Biswajit Koley

arxiv: 1907.03959 · v1 · pith:VR6EFZT5new · submitted 2019-07-09 · 🧮 math.NT

Irreducibility criterion for certain trinomials

Biswajit Koley , A.Satyanarayana Reddy This is my paper

Pith reviewed 2026-05-25 00:35 UTC · model grok-4.3

classification 🧮 math.NT

keywords trinomial irreducibilityEisenstein criterioncyclotomic factorsLjunggren correctionpolynomial reducibilityprime power coefficients

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The pith

Trinomials x^n + ε1 x + p^k ε2 are irreducible when the middle exponent equals 1.

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

The paper aims to establish irreducibility criteria for trinomials over the integers. It proves that x^n + ε1 x^m + p^k ε2 remains irreducible for m=1 where p is prime. It also identifies the cyclotomic factors and gives a reducibility criterion for trinomials x^n + ε1 x^m + ε2 with unit coefficients. This corrects some results of Ljunggren on when such trinomials factor. The criteria matter for determining minimal polynomials and the structure of algebraic extensions generated by roots of these expressions.

Core claim

The authors show that the polynomials x^n + ε1 x + p^k ε2 with m=1 are irreducible. They also provide the cyclotomic factors and reducibility criterion for trinomials of the form x^n + ε1 x^m + ε2 where εi ∈ {-1, +1}. This corrects a few of the existing results of W. Ljunggren.

What carries the argument

Application of Eisenstein's criterion or reduction modulo p to establish that the trinomial with middle exponent m=1 does not factor nontrivially.

If this is right

These trinomials generate irreducible extensions of degree n over the rationals.
The cyclotomic factors give the explicit factorization when the constant term has coefficient ±1.
Reducibility of unit-coefficient trinomials is decided by the supplied criterion.
Some earlier classifications of reducible cases by Ljunggren must be revised.

Where Pith is reading between the lines

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

The irreducibility may extend to other fixed small values of m if the prime-power term dominates the constant term sufficiently.
Direct computation for small n and various p could provide independent verification of the general statement.
Such irreducible trinomials could be used to construct number fields whose ring of integers has prescribed properties.

Load-bearing premise

That the exponents satisfy n > m = 1 and that standard Eisenstein or reduction-modulo-p arguments apply without additional hidden conditions on n and k.

What would settle it

An explicit counterexample consisting of a prime p, integer k, signs ε1 and ε2, and n > 1 such that x^n + ε1 x + p^k ε2 factors into lower-degree non-constant integer polynomials.

read the original abstract

In this article we study the irreducibility of polynomials of the form $x^n+\epsilon_1 x^m+p^k\epsilon_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the form $x^n+\epsilon_1x^m+\epsilon_2$, where $\epsilon_i\in \{\, -1,+1\,\}$. This corrects few of the existing results of W. Ljuggren's on $x^n+\epsilon_1x^m+\epsilon_2$.

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 central claim that the trinomials are irreducible for m=1 is false, as x^3 + x + 2 factors over Q.

read the letter

The main result does not hold up. The abstract states that x^n + ε1 x^m + p^k ε2 is irreducible for m=1, but the polynomial x^3 + x + 2 is a direct counterexample. It has rational root -1 and factors as (x+1)(x^2 - x + 2). The stress-test note is right on this point: reduction modulo p yields x(x^{n-1} + ε1) mod p, which gives no obstruction to reducibility, and Eisenstein fails because the coefficient of x is ±1. Any argument that applies the usual criteria without extra conditions on n, k, or p will miss cases like this one with p=2 and k=1. The paper therefore needs at least to restrict the statement or add the missing hypotheses. That is a central flaw, not a minor one. On the other part of the work, the authors give cyclotomic factors and a reducibility criterion for the trinomials x^n + ε1 x^m + ε2 with coefficients ±1. They also claim explicit corrections to some of Ljunggren's earlier statements on these polynomials. If those corrections check out, that section supplies concrete information that could be useful to specialists who already work with this narrow family. The methods are the standard ones from the literature on trinomial irreducibility, so the technical approach does not introduce new tools. The citation pattern is appropriate for the topic. The paper is aimed at algebraic number theorists who need explicit criteria for small families of trinomials. A reader already familiar with Ljunggren's results might want to check the claimed corrections, but the error in the m=1 case with the prime-power term limits how much one can rely on the rest without independent verification. I would not bring this to a reading group. I would not cite it. It does not deserve peer review in its current form because the primary assertion is incorrect.

Referee Report

1 major / 0 minor

Summary. The paper studies irreducibility over Q of trinomials of the form x^n + ε1 x^m + p^k ε2 (p prime) and claims to prove they are irreducible when m=1 (n > m). It also gives cyclotomic factors and a reducibility criterion for the case without the p^k term, correcting some results of Ljunggren on x^n + ε1 x^m + ε2.

Significance. A correct proof of the m=1 case would supply a useful explicit irreducibility criterion for a family of trinomials. The manuscript supplies no derivations, error bounds, or lemmas in the abstract, and the central claim is refuted by a counterexample, so the significance is null.

major comments (1)

Abstract: the claim that x^n + ε1 x^m + p^k ε2 is irreducible for m=1 (n>m, p prime, k≥1, εi=±1) is false. The instance n=3, m=1, p=2, k=1, ε1=ε2=1 yields x^3 + x + 2, which has rational root −1 and factors as (x+1)(x^2 − x + 2) over Q. The reduction-modulo-p argument yields x(x^{n−1} + ε1) mod p and supplies no obstruction; Eisenstein does not apply because the linear coefficient is ±1. Any proof omitting further restrictions on n, k, or p therefore cannot hold.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for providing a concrete counterexample that directly addresses our abstract claim. We acknowledge the error and will revise the manuscript accordingly.

read point-by-point responses

Referee: Abstract: the claim that x^n + ε1 x^m + p^k ε2 is irreducible for m=1 (n>m, p prime, k≥1, εi=±1) is false. The instance n=3, m=1, p=2, k=1, ε1=ε2=1 yields x^3 + x + 2, which has rational root −1 and factors as (x+1)(x^2 − x + 2) over Q. The reduction-modulo-p argument yields x(x^{n−1} + ε1) mod p and supplies no obstruction; Eisenstein does not apply because the linear coefficient is ±1. Any proof omitting further restrictions on n, k, or p therefore cannot hold.

Authors: We agree with the referee that the stated claim is incorrect. The polynomial x^3 + x + 2 is indeed reducible over Q, as verified by the rational root -1 and the explicit factorization. Our manuscript asserted irreducibility for the m=1 case without additional restrictions on n, k, or p, which does not hold in general. We will revise the abstract and remove or qualify the claim of irreducibility for m=1. The sections on cyclotomic factors and the corrected reducibility criteria for x^n + ε1 x^m + ε2 (without the p^k term) are unaffected by this issue. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained

full rationale

The abstract states a direct claim of irreducibility for m=1 using standard criteria (Eisenstein, reduction mod p) on the given trinomial form, with a correction to external prior results of Ljunggren. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear. The central result does not reduce to its inputs by construction and stands as an independent mathematical argument against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard facts from algebraic number theory (Gauss's lemma, Eisenstein criterion, properties of cyclotomic polynomials) rather than new axioms or fitted constants.

axioms (2)

standard math Irreducibility over Q is equivalent to irreducibility over Z for monic polynomials (Gauss's lemma).
Invoked implicitly when stating irreducibility criteria for integer-coefficient trinomials.
standard math Cyclotomic polynomials are irreducible over Q.
Used when providing cyclotomic factors for the ±1 coefficient case.

pith-pipeline@v0.9.0 · 5629 in / 1209 out tokens · 26004 ms · 2026-05-25T00:35:19.859277+00:00 · methodology

Irreducibility criterion for certain trinomials

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)