On $p$-th cyclotomic field and cyclotomic matrices involving Jacobi sums

Abstract

Inspired by Weil's classical result on the zeta function of projective Fermat curve defined over a finite field, in this paper, we investigate some arithmetic properties of the cyclotomic matrix $$\det\left[J_p(\chi^{ki},\chi^{kj})\right]_{1\le i,j\le n-1},$$ where $p\ge3$ is a prime, $1\le k<p-1$ is a divisor of $p-1$ with $p-1=kn$, $\chi$ is a generator of the group of all multiplicative characters of $\mathbb{F}_p$ and $J_p(\chi^{ki},\chi^{kj})$ is the Jacobi sum. For example, let $\zeta_p\in\mathbb{C}$ be a primitive $p$-th root of unity and $P_k(T)$ be the minimal polynomial of the algebraic integer $$\theta_k=\sum_{x\in\mathbb{F}_p,x^k=1}\zeta_p^x$$ over $\mathbb{Q}$. Then we prove that $$\det \left[J_p(\chi^{ki},\chi^{kj})\right]_{1\le i,j\le n-1}=(-1)^{\frac{(k+1)(n^2-n)}{2}}\cdot n^{n-2}\cdot x_p(k),$$ where $x_p(k)$ is the coefficient of $T$ in $P_k(T)$.