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

TL;DR

This paper explores the arithmetic properties of cyclotomic matrices constructed from Jacobi sums over finite fields, revealing explicit determinant formulas linked to algebraic integers and minimal polynomials.

Contribution

It provides a novel explicit formula for the determinant of cyclotomic matrices involving Jacobi sums, connecting them to algebraic integers and minimal polynomials.

Findings

Determinant expressed in terms of algebraic integer coefficients

Explicit formula involving primitive roots of unity

Connection between matrix determinants and minimal polynomials

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 $P_k(T)$.