Generic extensions and generic Polynomials for multiplicative groups

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 $S\mathrm{/}R$, 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.