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Regular version of the inverse Galois problem for skew fields

Antonin Assoun

TL;DR

This work generalizes the regular inverse Galois problem to arbitrary fields by defining InvReg$(k)$ and the regular inverse Galois problem for regular extensions $K/k$, and then extends the framework to skew fields. It shows that skew fraction fields $h(t,\sigma)$ and $h(t,\sigma,\delta)$ belong to InvReg$(h)$ under suitable center/ampleness hypotheses, and it generalizes BeH's scalar-extension method to cases where $\sigma$ is locally of finite order, enabling noncommutative regular extensions beyond the finite-order case. A noncommutative Artin lemma and an inductive-limit tower formalism are developed to transfer Galois-group data through towers of twisted fraction fields, which are then applied to construct Galois extensions with prescribed groups over skew-field towers while preserving regularity. The results culminate in Theorem 1, showing the existence of regular, noncommutative extensions in InvReg$(h)$ not realizable as $h(t,\sigma)/h$ with finite-order $\sigma$, highlighting new realizations of finite groups in the noncommutative setting and broadening the scope of the regular inverse Galois problem.

Abstract

In order to extend the study of the regular version of the regular inverse Galois problem to skew fields, we generalize the definition of regular field extensions for commutative fields to the case of arbitrary fields. We then propose a general version of the inverse Galois property and show that for a field k, the study of the class InvReg(k) of non-trivial regular extensions of k satisfying this property constitutes a natural generalization of the classical regular inverse Galois problem. Next, we use recent results from Behajaina, Deschamps, and Legrand on the inverse Galois problem in the noncommutative setting to show that certain skew fraction fields h(t, σ, δ) belong to the class InvReg(h). Finally, we generalize Behajaina's method to construct extensions belonging to InvReg(h) that are not of the form h(t, σ) with σ an automorphism of finite order.

Regular version of the inverse Galois problem for skew fields

TL;DR

This work generalizes the regular inverse Galois problem to arbitrary fields by defining InvReg and the regular inverse Galois problem for regular extensions , and then extends the framework to skew fields. It shows that skew fraction fields and belong to InvReg under suitable center/ampleness hypotheses, and it generalizes BeH's scalar-extension method to cases where is locally of finite order, enabling noncommutative regular extensions beyond the finite-order case. A noncommutative Artin lemma and an inductive-limit tower formalism are developed to transfer Galois-group data through towers of twisted fraction fields, which are then applied to construct Galois extensions with prescribed groups over skew-field towers while preserving regularity. The results culminate in Theorem 1, showing the existence of regular, noncommutative extensions in InvReg not realizable as with finite-order , highlighting new realizations of finite groups in the noncommutative setting and broadening the scope of the regular inverse Galois problem.

Abstract

In order to extend the study of the regular version of the regular inverse Galois problem to skew fields, we generalize the definition of regular field extensions for commutative fields to the case of arbitrary fields. We then propose a general version of the inverse Galois property and show that for a field k, the study of the class InvReg(k) of non-trivial regular extensions of k satisfying this property constitutes a natural generalization of the classical regular inverse Galois problem. Next, we use recent results from Behajaina, Deschamps, and Legrand on the inverse Galois problem in the noncommutative setting to show that certain skew fraction fields h(t, σ, δ) belong to the class InvReg(h). Finally, we generalize Behajaina's method to construct extensions belonging to InvReg(h) that are not of the form h(t, σ) with σ an automorphism of finite order.

Paper Structure

This paper contains 16 sections, 19 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Regularity of Field Extensions and Regular Inverse Galois Problem
  3. Regular Extensions
  4. Existentially Closed Extensions
  5. Skew Series Fields
  6. Regular Form of the Inverse Galois Problem
  7. Interpretation of the work of Behajaina, Deschamps, and Legrand
  8. Permanence of the Galois group in towers of Galois extensions
  9. Generalized Artin Lemma
  10. Isomorphisms of Towers of Galois Extensions
  11. Skew Fraction Fields Extensions
  12. Application to the Inverse Galois Problem
  13. Tower with a Fixed Galois Group
  14. Towers Satisfying $\mathrm{IGP}$
  15. Regularity of the towers
  16. ...and 1 more sections

Key Result

Theorem 1

Let $h/k$ be an algebraic outer Galois extension of fields such that $\mathrm{Gal}(h/k)$ is an infinite product of cyclic groups of coprime orders. If $[Z(h):Z(k)] = +\infty$, and $Z(k)$ contains an ample field, then the class $\mathrm{InvReg}(h)$ contains extensions of $h$ that are non-commutative

Theorems & Definitions (22)

  • Definition 1: $\mathbf{RIGP}_{K/k}$ Regular Inverse Galois Problem for a regular extension $K/k$
  • Theorem 1
  • Definition 2
  • Proposition 1
  • Theorem 2
  • Lemma 1
  • Proposition 2
  • Definition 3
  • Proposition 3
  • Theorem 3
  • ...and 12 more