Frobenius matrices and a variant of Zolotarev's theorem

TL;DR

This paper generalizes Zolotarev's theorem to finite-dimensional vector spaces over finite fields using matrix theory, expanding its applicability beyond classical settings.

Contribution

It introduces a novel generalization of Zolotarev's theorem applicable to arbitrary finite-dimensional vector spaces over finite fields.

Findings

Extended Zolotarev's theorem to finite fields

Developed new matrix-based proof techniques

Broadened the theorem's applicability to finite-dimensional spaces

Abstract

In this paper, with the help of the theory of matrices and finite fields we generalize Zolotarev's theorem to an arbitrary finite dimensional vector space over $\mathbb{F}_q$, where $\mathbb{F}_q$ denotes the finite field with $q$ elements.