On a polynomial involving roots of unity and its applications

Abstract

Let [Formula: see text] be a prime. Gauss first introduced the polynomial [Formula: see text] where [Formula: see text] and [Formula: see text] varies over all quadratic residues modulo p and [Formula: see text]. Later Dirichlet investigated this polynomial and used this to solve problems involving the Pell equations. Recently, Sun studied some trigonometric identities involving this polynomial. In this paper, we generalize their results. As applications of our result, we extend Chowla’s result on the congruence concerning the fundamental unit of [Formula: see text] and give an equivalent form of the extended Ankeny et al. conjecture.