On a polynomial involving quadratic residues modulo primes

Let $p$ be an odd prime, and define $$G_p(x)=\prod _{k=1}^{(p-1)\mathrm{/}2}(x-e^{2\pi i{k}^2\mathrm{/}p})\mathrm{.}$$ 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 $$G_p(\zeta )=\{\begin{array}{ll}(-1{)}^{\mathrm{\mid }\{1\le k\le \frac{p+9}{10}:(\frac{k}{p})=-1\}\mathrm{\mid }}& \text{if }p\equiv 21(\mathrm{m}\mathrm{o}\mathrm{d}40),\\ (-1{)}^{\mathrm{\mid }\{1\le k\le \frac{p+1}{10}:(\frac{k}{p})=-1\}\mathrm{\mid }}{\zeta }^2& \text{if }p\equiv 29(\mathrm{m}\mathrm{o}\mathrm{d}40),\end{array}$$ where $(\frac{k}{p})$ denotes the Legendre symbol.