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Artin-Schreier quandles of involutions in absolute Galois groups

Markus Szymik

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.

Artin-Schreier quandles of involutions in absolute Galois groups

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.
Paper Structure (5 sections, 19 theorems, 21 equations, 1 figure)

This paper contains 5 sections, 19 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Involutions and quandles
  3. Artin--Schreier quandles
  4. Number fields
  5. Laurent series and other examples

Key Result

Theorem 1.1

The Artin--Schreier quandle $\mathrm{AS}(\mathbb{Q})$ of the rational number field $\mathbb{Q}$ is a free pro-finite involutory quandle. A Cantor space of involutions inside the absolute Galois group $\mathop{\mathrm{Gal}}\nolimits(\mathbb{Q})$ gives a basis. A similar statement holds for all formal

Figures (1)

  • Figure 1: The rack axiom \ref{['eq:rack_axiom']} for symmetric spaces

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Proposition 2.10
  • ...and 47 more