[Paper Review] Le probl\\`eme inverse de Galois sur les corps des fractions tordus \\`a ind\\'etermin\\'ee centrale

Bruno Deschamps, François Legrand|arXiv (Cornell University)|Mar 26, 2019

Algebraic Geometry and Number Theory21 references13 citations

TL;DR

This paper establishes an equivalence between the inverse Galois problem over a central skew field $H$ of finite dimension over its center $k$, and a constrained variant of the inverse Galois problem over $k$ involving the norm form of $H/k$. The key result shows that if $k$ contains an ample field, then every finite group arises as a Galois group over the twisted rational function field $H(t)$, resolving the inverse Galois problem in this noncommutative setting via a polynomial constraint on the base field.

ABSTRACT

In this article, we show that the Inverse Galois Problem over a skew field $H$ of finite dimension over its center $k$ is equivalent to a variant of the Inverse Galois Problem over $k$ involving a polynomial constraint. As an application, we show that if $k$ contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field $H(t)$ of rational fractions with central indeterminate.

Motivation & Objective

• To establish a precise equivalence between the inverse Galois problem over a central skew field $H$ and a constrained variant over its center $k$.
• To extend the inverse Galois problem to noncommutative settings, particularly twisted rational function fields $H(t)$ with central indeterminate.
• To show that the existence of an ample field inside $k$ ensures a positive solution to the inverse Galois problem over $H(t)$.
• To clarify the role of the norm form in controlling linear disjointness between extensions, enabling Galois realization over noncommutative fields.
• To generalize classical Galois theory techniques to skew fields and twisted rational function fields, where standard algebraic closure concepts fail.

Proposed method

• Define the norm form $\mathcal{F} = P_H$ associated with the reduced norm of $H/k$, which encodes the linear disjointness condition between $H/k$ and a Galois extension $L/k$.
• Prove that $H/k$ and $L/k$ are linearly disjoint if and only if $\mathcal{F}$ has no nontrivial zero in $L$, linking the Galois realization over $H$ to the vanishing of $\mathcal{F}$ over $L$.
• Establish the equivalence $\text{PIG}_{\mathcal{F}\text{C}}^k \Longleftrightarrow \text{PIG}_H$ via a duality argument using the structure of finite-dimensional central simple algebras.
• Use the existence of $n+1$ linearly disjoint Galois extensions of degree $G^{n+1}$ to ensure at least one such extension is disjoint from $H$, leveraging the number of intermediate fields.
• Construct the twisted rational function field $H(t)$ as the quotient field of the twisted polynomial ring $H[t]$ with central indeterminate, using Ore's construction.
• Apply the equivalence result to show that if $k$ contains an ample field, then $\text{PIG}_{H(t)}$ has a positive solution, due to the existence of suitable $L/k$ with trivial $\mathcal{F}$-zeros.

Experimental results

Research questions

• RQ1 Is the inverse Galois problem over a finite-dimensional central skew field $H$ over $k$ equivalent to a constrained inverse Galois problem over $k$ involving the norm form of $H/k$?
• RQ2 Can the inverse Galois problem be solved over the twisted rational function field $H(t)$ when $k$ contains an ample field?
• RQ3 What role does the norm form $P_H$ play in ensuring linear disjointness between $H/k$ and a Galois extension $L/k$?
• RQ4 Why does the implication $\text{PIG}_H \Rightarrow \text{PIG}_{\mathcal{F}\text{C}}^k$ fail in general, yet hold in the noncommutative setting?
• RQ5 How can classical Galois theory be adapted to noncommutative fields, particularly in the context of twisted rational function fields?

Key findings

• The inverse Galois problem over a finite-dimensional central skew field $H$ over $k$ is equivalent to the constrained inverse Galois problem over $k$ involving the norm form $P_H$ of $H/k$, establishing a precise algebraic bridge between the two.
• The equivalence $\text{PIG}_{\mathcal{F}\text{C}}^k \Longleftrightarrow \text{PIG}_H$ holds due to the interplay between linear disjointness and the vanishing of the norm form over extension fields.
• If $k$ contains an ample field, then every finite group arises as a Galois group over $H(t)$, the twisted rational function field with central indeterminate, providing a positive solution to the inverse Galois problem in this noncommutative setting.
• The construction of $H(t)$ as the quotient field of the twisted polynomial ring $H[t]$ with central $t$ ensures that all elements are of the form $p(t)q(t)^{-1}$, preserving the noncommutative structure.
• The proof relies on the existence of sufficiently many linearly disjoint Galois extensions of $k$ to ensure that at least one is disjoint from $H$, leveraging the number of intermediate fields in $H/k$.
• The result demonstrates that the inverse Galois problem is more tractable in noncommutative settings when the base field $k$ has strong model-theoretic properties such as being ample.