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arxiv: 2311.14953 · v1 · pith:UMMFENL7new · submitted 2023-11-25 · 🧮 math.NT

A note on Galois groups of linearized polynomials

Peter M\"uller This is my paper

Pith reviewed 2026-05-24 06:29 UTC · model grok-4.3

classification 🧮 math.NT
keywords Galois groupslinearized polynomialsfunction fieldsHensel's lemmafinite fieldsgeneral linear groupnumber theory
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The pith

The Galois group of L(X) minus t over F_q(t) equals the full GL_n(q) for any prime power q when n is an odd prime and L is not X to the q^n.

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

The paper proves that for a monic q-linearized polynomial L of degree q^n over the finite field F_q, with n an odd prime, the Galois group of the extension defined by L(X) equals t over the rational functions F_q(t) is the general linear group GL_n(q), except when L is simply the monomial X to the power q^n. Earlier work had established this for odd q, leaving even q open, and this note confirms the conjecture that the result holds in the even case as well. The argument applies Hensel's lemma directly in the function-field setting to obtain a single proof that covers every prime power q without splitting into cases. A reader would care because this determines the full symmetry group of these particular extensions, which control the factorization behavior and splitting of linearized polynomials in function fields.

Core claim

Let L(X) be a monic q-linearized polynomial over F_q of degree q^n, where n is an odd prime. The Galois group of L(X) over X minus t in the rational function field F_q(t) is GL_n(q) unless L(X) equals X to the power q^n. We settle the conjecture that this holds for even q as well. In fact we use Hensel's lemma to give a unified proof for all prime powers q.

What carries the argument

Hensel's lemma applied to the Galois extension of the function field F_q(t) to control the image of the Galois group inside GL_n(q).

If this is right

  • The full group GL_n(q) acts as Galois group for every such linearized polynomial when q is even.
  • No separate case analysis by the parity of q is needed.
  • The conjecture of Gow and McGuire is now proved in complete generality for odd prime n.

Where Pith is reading between the lines

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

  • The same lifting technique might extend the result to certain composite values of n where the ramification can still be controlled.
  • It would be natural to test whether the method adapts to other families of polynomials whose Galois groups are expected to be large linear groups over function fields.
  • Small explicit cases with even q and small odd prime n admit direct computational verification of the group order or generators as an independent check.

Load-bearing premise

The argument requires n to be an odd prime so that the ramification or the action of the group can be controlled sufficiently when Hensel's lemma is applied in the completion at a suitable place.

What would settle it

A concrete counterexample would be an explicit even q, odd prime n, and monic q-linearized L of degree q^n not equal to X^{q^n} for which the splitting field of L(X) minus t over F_q(t) has Galois group strictly smaller than GL_n(q).

read the original abstract

Let $L(X)$ be a monic $q$-linearized polynomial over $F_q$ of degree $q^n$, where $n$ is an odd prime. Recently Gow and McGuire showed that the Galois group of $L(X)/X-t$ over the field of rational functions $F_q(t)$ is $GL_n(q)$ unless $L(X)=X^{q^n}$. The case of even $q$ remained open, but it was conjectured that the result holds too and partial results were given. In this note we settle this conjecture. In fact we use Hensel's Lemma to give a unified proof for all prime powers $q$.

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 proves that if L(X) is a monic q-linearized polynomial of degree q^n over F_q with n an odd prime, then the Galois group of the extension defined by L(X) - t over F_q(t) equals GL_n(q) unless L(X) = X^{q^n}. It supplies a unified argument via Hensel's lemma that covers all prime powers q, including the previously open even-q case, thereby settling the conjecture left open by Gow and McGuire.

Significance. The result completes the classification of these Galois groups in the stated setting. The use of a standard Hensel's-lemma lifting argument to obtain uniformity across all q is a methodological strength; the proof is parameter-free once the reduction step is fixed and builds directly on cited prior results without circularity.

minor comments (3)
  1. [Section 3] The choice of place and the explicit verification that the derivative condition holds after reduction (for even q) should be stated in a single numbered lemma or proposition rather than distributed across the proof paragraphs.
  2. [Introduction] A short remark clarifying why the oddness of n is retained in the statement (even though the lifting works for all q) would help readers compare with the Gow-McGuire result.
  3. [Section 2] The notation for the completion and residue field in the function-field setting is introduced without a dedicated sentence; a one-line definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so there are no specific points requiring a point-by-point response.

Circularity Check

0 steps flagged

No circularity: proof applies standard Hensel's lemma to extend independent prior result

full rationale

The derivation begins from the Gow-McGuire theorem (distinct authors) establishing the Galois group equals GL_n(q) when q is odd, then invokes the classical Hensel's lemma to lift the reduction modulo a place in F_q(t) and obtain the same group for even q. No equation or claim reduces a fitted parameter to a prediction, no ansatz is smuggled via self-citation, and the central statement is not defined in terms of itself. The argument is self-contained against external benchmarks (Hensel's lemma and the cited Galois-group computation) and does not rely on any load-bearing self-reference.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from algebra (Hensel's lemma) and the prior Galois-group computations of Gow and McGuire; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Hensel's lemma holds in the relevant completion or local ring arising from the function field F_q(t).
    Hensel's lemma is invoked to lift solutions; this is a classical theorem whose hypotheses must be verified in the paper's setting.
  • domain assumption The Galois group computations and group-theoretic facts from Gow and McGuire hold for the odd-q case and provide the base for the unified argument.
    The note explicitly builds on the cited prior result.

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discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 1 internal anchor

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    Cameron, P.J., Kantor, W.M.: 2–transitive and antiflag transitive collineation groups of finite projective spaces. J. Algebra 60, 384–422 (1979)

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    Kantor, W.M.: Linear groups containing a Singer cycle. J. Algebra 62(1), 232–234 (1980)

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    Antiflag Transitive Collineation Groups

    Kantor, W.M.: Antiflag transitive collineation groups (2018). ArXiv:1806.02203

    work page internal anchor Pith review Pith/arXiv arXiv 2018

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    Turnwald, G.: On Schur’s conjecture. J. Austral. Math. Soc. Se r. A 58, 312–357 (1995) 5

    work page 1995