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arxiv: 2403.07545 · v2 · submitted 2024-03-12 · 🧮 math.NT · math.GR· math.GT· math.QA

Artin-Schreier quandles of involutions in absolute Galois groups

Markus Szymik This is my paper

Pith reviewed 2026-05-24 03:17 UTC · model grok-4.3

classification 🧮 math.NT math.GRmath.GTmath.QA
keywords Artin-Schreier quandleabsolute Galois groupreal spectrumformally real fieldsnumber fieldsLaurent series fieldsinvolutions
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The pith

The Artin-Schreier quandle is a new invariant of fields that refines their real spectrum and incorporates data from involutions in the absolute Galois group.

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

The paper defines the Artin-Schreier quandle as a functorial invariant attached to any field. This object refines the real spectrum by adding information drawn from the absolute Galois group. For formally real number fields the quandle is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields the quandle is determined by the corresponding quandle of the coefficient field. The paper also exhibits examples in which relations appear among the generators.

Core claim

The Artin-Schreier quandle of a field is introduced as an invariant that refines the real spectrum and is related to the absolute Galois group. For formally real number fields it is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields it is computed in terms of the Artin-Schreier quandle of the coefficient field, and other examples demonstrate that relations exist in general.

What carries the argument

The Artin-Schreier quandle constructed from involutions in the absolute Galois group, which carries the Galois-theoretic data and refines the real spectrum.

If this is right

  • Formally real number fields have Artin-Schreier quandles that are freely generated by a Cantor space.
  • The Artin-Schreier quandle of a Laurent series field is determined by the quandle of its coefficient field.
  • In general the Artin-Schreier quandle of a field exhibits relations among its generators.
  • The quandle distinguishes fields at a level finer than the real spectrum alone.

Where Pith is reading between the lines

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

  • The free generation property for number fields suggests that the quandle encodes independent choices compatible with the real spectrum.
  • The appearance of relations in other cases indicates that the quandle can detect dependencies that the real spectrum does not record.

Load-bearing premise

The Artin-Schreier quandle is a well-defined functorial invariant on fields that genuinely refines the real spectrum while carrying Galois-theoretic information.

What would settle it

An explicit computation of the Artin-Schreier quandle for a concrete formally real number field that fails to be freely generated by a Cantor space of indeterminates would show the central claim does not hold.

Figures

Figures reproduced from arXiv: 2403.07545 by Markus Szymik.

Figure 1
Figure 1. Figure 1: The rack axiom (2.1) for symmetric spaces This particular rack is an involutory quandle. Bachmann [5, 6], building on earlier work of Hjelmslev, has used groups generated by involutions as a foundation for geometry in a way that is distinctly different from Coxeter’s better-known work on reflection groups. Examples 2.9. For any group G, the subset Inv(G) = {s ∈ G | s 2 = id ̸= s} (2.4) of involutions in G … view at source ↗
read the original abstract

We introduce a new invariant of fields that refines their real spectrum and is related to their absolute Galois group: the Artin-Schreier quandle. For formally real number fields, it is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields, we compute it in terms of the Artin-Schreier quandle of the coefficient field. This result and other examples show that, in general, there are relations.

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 / 0 minor

Summary. The manuscript introduces the Artin-Schreier quandle, constructed from involutions in the absolute Galois group of a field, as a new invariant that refines the real spectrum. It asserts that for formally real number fields this quandle is freely generated in its variety by a Cantor space of indeterminates. For Laurent series fields it gives an explicit computation in terms of the quandle of the coefficient field, together with examples indicating that relations appear in general.

Significance. If the constructions and computations hold, the work supplies a functorial Galois-theoretic refinement of the real spectrum with concrete structural results for two important classes of fields. The freeness statement for number fields would constitute a strong, explicit description, while the Laurent-series reduction provides a recursive tool for generating further examples.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments were listed in the report, so we have no points to address individually at this stage. We remain available to supply further details, examples, or clarifications on the Artin-Schreier quandle construction, the freeness result for formally real number fields, or the reduction for Laurent series fields if the referee or editor requests them.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the Artin-Schreier quandle as a new functorial invariant on fields that refines the real spectrum and connects to the absolute Galois group, then derives explicit computations for formally real number fields (freely generated by a Cantor space) and Laurent series fields (in terms of the coefficient field's quandle). These steps rest on the initial definition and standard Galois-theoretic constructions rather than any self-referential reduction, fitted parameters presented as predictions, or load-bearing self-citations. No equations or claims reduce by construction to their own inputs, and the derivation chain remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only: the central claim rests on the existence and functoriality of the newly defined Artin-Schreier quandle together with the algebraic variety in which it lives; no explicit free parameters, background axioms, or invented entities are stated in the abstract.

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Reference graph

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