Rank Duality of Circulant Matrices from Primitive Roots

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.