Rank Duality of Circulant Matrices from Primitive Roots

TL;DR

This paper explores the structure of circulant matrices built from primitive roots over finite fields, linking character sums to matrix properties and revealing new insights into their algebraic and combinatorial applications.

Contribution

It introduces a novel method connecting exponential sums to Jacobi sums, enhancing understanding of circulant matrices derived from primitive roots.

Findings

Explicit connections between character theory and matrix structures

Reduction of exponential sums to Jacobi sums

Insights into the algebraic properties of circulant matrices

Abstract

We investigate the construction of circulant matrices derived from primitive roots over finite fields. Our approach reduces exponential sums to Jacobi sums, thereby establishing explicit connections between character theory and matrix structures. The results provide new insights into the interaction between additive and multiplicative characters, and demonstrate how circulant configurations encode arithmetic information in a highly symmetric form. These findings contribute to a deeper understanding of structured matrices in finite fields and open further directions for applications in number theory and combinatorics.