[Paper Review] Galois groups over rational function fields over skew fields

Gil Alon, François Legrand+1 more|arXiv (Cornell University)|Feb 24, 2020

Algebraic Geometry and Number Theory7 references4 citations

TL;DR

This paper solves the Inverse Galois Problem over the rational function field $ H(X) $, where $ H $ is a skew field of finite dimension over its center $ k $, provided $ k $ contains an ample field. By proving that $ H(X) $ is isomorphic to a rational function field in $ n $ central variables over $ k $, the authors reduce the problem to a previously solved case, thereby establishing that every finite group arises as a Galois group over $ H(X) $.

ABSTRACT

Let $H$ be a skew field of finite dimension over its center $k$. We solve the Inverse Galois Problem over the field of fractions $H(X)$ of the ring of polynomial functions over $H$ in the variable $X$, if $k$ contains an ample field.

Motivation & Objective

• To extend the Inverse Galois Problem to rational function fields over skew fields, particularly $ H(X) $, where $ H $ is a finite-dimensional skew field over its center $ k $.
• To address the gap in inverse Galois theory for non-commutative settings by generalizing results from commutative function fields to skew field extensions.
• To establish that every finite group occurs as a Galois group over $ H(X) $, under the condition that the center $ k $ contains an ample field.
• To bridge the gap between two distinct polynomial constructions over skew fields: $ H[X] $ (non-central variable) and $ H_c[t_1,\dots,t_n] $ (central variables), by proving their isomorphism.
• To demonstrate that the Inverse Galois Problem over skew fields is an algebraic invariant under isomorphism, enabling transfer of results between isomorphic fields.

Proposed method

• Prove that the ring $ H[X] $ of polynomial functions over a skew field $ H $ in a non-central variable $ X $ is isomorphic to the polynomial ring $ H_c[t_1,\dots,t_n] $ in $ n $ central variables, where $ n = [H:k] $, using a result of Wilczynski.
• Establish that $ H[X] $ is a right Ore domain when $ k $ is infinite, ensuring the existence of a classical right field of fractions $ H(X) $.
• Use the isomorphism $ H(X) \cong H_c(t_1,\dots,t_n) $ to reduce the problem over $ H(X) $ to the known case over $ H_c(t_1,\dots,t_n) $, leveraging existing results from Deschamps and Legrand.
• Apply the algebraic invariance of the Inverse Galois Problem under isomorphism of skew fields (Proposition 3.1), showing that if a finite group arises as a Galois group over $ H_c(t_1,\dots,t_n) $, it also does so over $ H(X) $.
• Verify that the center of $ H_c(t_1,\dots,t_{n-1}) $ is $ k(t_1,\dots,t_{n-1}) $, which inherits the ample property from $ k $, allowing the application of [DL19, Théorème B].
• Leverage the fact that $ H_c(t_1,\dots,t_n) $ is a skew field extension of finite dimension over its center, ensuring the applicability of the foundational result from Deschamps and Legrand.

Experimental results

Research questions

• RQ1 Can every finite group be realized as a Galois group over the rational function field $ H(X) $, where $ H $ is a skew field of finite dimension over its center $ k $, if $ k $ contains an ample field?
• RQ2 Is the ring $ H[X] $ of polynomial functions over a skew field $ H $ in a non-central variable isomorphic to a polynomial ring in $ n $ central variables over $ H $, where $ n = [H:k] $?
• RQ3 Does the Inverse Galois Problem over skew fields depend only on the isomorphism class of the base skew field, or can it be transferred across isomorphic fields?
• RQ4 What is the structure of the field of fractions $ H(X) $, and how does it relate to the field of rational functions in central variables over the center $ k $?
• RQ5 Under what conditions on the center $ k $ of a finite-dimensional skew field $ H $ does the Inverse Galois Problem have a positive solution over $ H(X) $?

Key findings

• The field of fractions $ H(X) $ of the polynomial ring $ H[X] $ in a non-central variable $ X $ is isomorphic to the rational function field $ H_c(t_1,\dots,t_n) $ in $ n $ central variables, where $ n = [H:k] $, provided $ k $ is infinite.
• The Inverse Galois Problem has a positive solution over $ H(X) $ whenever the center $ k $ of the skew field $ H $ contains an ample field, meaning every finite group arises as a Galois group over $ H(X) $.
• The isomorphism $ H(X) \cong H_c(t_1,\dots,t_n) $ allows the transfer of results from the central variable case to the non-central case, thereby solving the problem via reduction to [DL19, Théorème B].
• The Inverse Galois Problem is invariant under isomorphism of skew fields: if a finite group $ G $ is realizable over one skew field, it is realizable over any isomorphic skew field.
• The result applies to fundamental cases such as $ \mathbb{H}(X) $, where $ \mathbb{H} $ is the skew field of Hamilton’s quaternions, since $ \mathbb{R} $ (the center of $ \mathbb{H} $) contains ample fields like $ \mathbb{Q}^{\rm{tr}} $.
• Additional cases are covered by the remark: for example, if $ k $ is infinite, every abelian group, symmetric group $ S_m $ ($ m \geq 3 $), alternating group $ A_m $ ($ m \geq 4 $) in characteristic zero, or solvable group with $ n \geq 2 $ in positive characteristic, can be realized as Galois groups over $ H(X) $.