Generic extensions and generic Polynomials for multiplicative groups

TL;DR

This paper constructs explicit generic Galois extensions and polynomials for multiplicative groups of finite-dimensional algebras over finite fields, advancing the understanding of their algebraic and Galois-theoretic properties.

Contribution

It provides explicit constructions of generic Galois extensions and polynomials for multiplicative groups of finite-dimensional algebras over finite fields.

Findings

Explicit generic Galois $G$-extension $S/R$ constructed

Explicit generic polynomial for $G$ with $ ext{dim}_{f F_q}( ext{A})$ parameters

Advances in understanding Galois theory of multiplicative groups

Abstract

Let $\mathcal{A}$ be a finite-dimensional algebra over a finite field $\mathbf{F}_q$ and let $G=\mathcal{A}^\times$ be the multiplicative group of $\mathcal{A}$. In this paper, we construct explicitly a generic Galois $G$-extension $S/R$, where $R$ is a localized polynomial ring over $\mathbf{F}_q$, and an explicit generic polynomial for $G$ in $\dim_{\mathbf{F}_q}(\mathcal{A})$ parameters.