Frobenius matrices and a variant of Zolotarev's theorem

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.

Our result is not new, which was obtained by Prof. P.L. Clark and Prof. A. Brunyate in the paper Extending the Zolotarev-Frobenius approach to quadratic reciprocity (pubished in the Ramanujan journal)