On cyclotomic matrices involving Gauss sums over finite fields
Hailiang Wu, Jie Li, Liyuan Wang, Chi Hoi Yip · Proceedings of the American Mathematical Society · 2024
Inspired by the works of L. Carlitz [Acta Arith. 5 (1959), pp. 293–308] and Z.-W. Sun [Finite Fields Appl. 56 (2019), pp. 285–307] on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let q = p n q=p^n be an odd prime power with p p prime and n ∈ Z + n\in \mathbb {Z}^+ . Let ζ p = e 2 π i / p \zeta _p=e^{2\pi \mathbf {i}/p} and let χ \chi be a generator of the group of all multiplicative characters of the finite field F q \mathbb {F}_q . For the Gauss sum G q ( χ r ) = ∑ x ∈ F q χ r ( x ) ζ p T r F q