On a polynomial involving roots of unity and its applications
Hai-Liang Wu, Yue-Feng She
TL;DR
The paper investigates $S_n(i)$ for squarefree $n>3$ with $n\equiv3\,\mathrm{mod}\,4$, defined via roots of unity and a Jacobi symbol constraint. It combines class-number theory, Gauss sum arguments, and Galois automorphisms to derive explicit closed forms of $S_n(i)$ in both composite and prime cases, expressing them in terms of the fundamental unit $\varepsilon_{4n}$ and the class number $h(4n)$, and obtaining Pell-type relations for the coefficients. The main contributions are the explicit formulas $S_n(i)=(-1)^{\phi(n)/8+\alpha(n)}\varepsilon_{4n}^{-h(4n)/2}$ for non-prime $n$ and $S_p(i)=(-1)^{\beta(p)}\varepsilon_{4p}^{-h(4p)/2}\frac{1+i(-1)^{(p+1)/4}}{\sqrt{2}}$ for prime $p$, along with the implication that $(i-(-1)^{(p+1)/4})S_p(i)=a_p+b_p\sqrt{p}$ with $a_p^2-pb_p^2=2\cdot(2/p)$. These results extend S. Chowla's congruence on fundamental units and yield an equivalent form of the Extended Ankeny-Artin-Chowla conjecture when linked to the condition $p\nmid b_p$. The work thus connects cyclotomic-root polynomials with arithmetic of real and imaginary quadratic fields and their class numbers.
Abstract
Let $p>3$ be a prime. Gauss first introduced the polynomial $S_p(x)=\prod_{c}(x-ζ_p^c),$ where $0<c<p$ and $c$ varies over all quadratic residues modulo $p$ and $ζ_p=e^{2πi/p}$. Later Dirichlet investigated this polynomial and used this to solve the problems involving the Pell equations. Recently, Z.-W Sun studied some trigonometric identities involving this polynomial. In this paper, we generalized their results. As applications of our result, we extend S. Chowla's result on the congruence concerning the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and give an equivalent form of the extended Ankeny-Artin-Chowla conjecture.