On a polynomial involving quadratic residues modulo primes

TL;DR

This paper investigates a polynomial involving quadratic residues modulo primes, analyzing its values at roots of unity using Galois theory, and confirms specific conjectures about its behavior depending on prime congruences.

Contribution

It provides a rigorous analysis of the polynomial's values at roots of unity and confirms conjectures relating these values to Legendre symbols and prime congruences.

Findings

Explicit formulas for G_p(ζ) at primitive tenth roots of unity.

Confirmation of conjectures relating G_p(ζ) to Legendre symbols.

Results depend on the residue class of p modulo 40.

Abstract

Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2\pi i k^2/p}\right).$$ In this paper we study values of $G_p(x)$ at roots of unity via Galois theory, and confirm some previous conjectures. For example, for any primitive tenth root $\zeta$ of unity, we prove that $$G_p(\zeta)=\begin{cases}(-1)^{|\{1\le k\le \frac {p+9}{10}:\ (\frac kp)=-1\}|} &\text{if}\ p\equiv21\pmod{40}, \\(-1)^{|\{1\le k\le\frac {p+1}{10}:\ (\frac kp)=-1\}|}\zeta^{2}&\text{if}\ p\equiv 29\pmod{40}, \end{cases}$$ where $(\frac kp)$ denotes the Legendre symbol.