The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups

Abstract: In this paper, by using the arithmetic properties of the Gauss periods and character sums over cyclic groups, we study the cyclotomic matrix $$A_k(χ)=\left[G_N(χ^{ki+ki})\right]_{0\le i,j\le \varphi(N)/k-1},$$ where $N=p^m$ is a prime power, $\varphi(\cdot)$ is the Euler totient function, $k$ is a divisor of $\varphi(N)$, $χ$ is a generator of character group $\widehat{(\mathbb{Z}/N\mathbb{Z})^{\times}}$, and $$G_N(χ^{ki+kj})=\sum_{x\in\mathbb{Z}/N\mathbb{Z}}χ^{ki+kj}(x)e^{2πix/N}$$ is the Gauss sum over $\mathbb{Z}/N\mathbb{Z}$.