On a polynomial involving quadratic residues modulo primes
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909032732688384 |
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| author | Sun, Zhi-Wei |
| author_facet | Sun, Zhi-Wei |
| contents | Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2πi k^2/p}\right).$$ 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 $ζ$ of unity, we prove that $$G_p(ζ)=\begin{cases}(-1)^{|\{1\le k\le \frac {p+9}{10}:\ (\frac kp)=-1\}|} &\text{if}\ p\equiv21\pmod{40}, \\(-1)^{|\{1\le k\le\frac {p+1}{10}:\ (\frac kp)=-1\}|}ζ^{2}&\text{if}\ p\equiv 29\pmod{40}, \end{cases}$$ where $(\frac kp)$ denotes the Legendre symbol. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05200 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a polynomial involving quadratic residues modulo primes Sun, Zhi-Wei Number Theory 11A15, 11C08, 11R11 Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2πi k^2/p}\right).$$ 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 $ζ$ of unity, we prove that $$G_p(ζ)=\begin{cases}(-1)^{|\{1\le k\le \frac {p+9}{10}:\ (\frac kp)=-1\}|} &\text{if}\ p\equiv21\pmod{40}, \\(-1)^{|\{1\le k\le\frac {p+1}{10}:\ (\frac kp)=-1\}|}ζ^{2}&\text{if}\ p\equiv 29\pmod{40}, \end{cases}$$ where $(\frac kp)$ denotes the Legendre symbol. |
| title | On a polynomial involving quadratic residues modulo primes |
| topic | Number Theory 11A15, 11C08, 11R11 |
| url | https://arxiv.org/abs/2605.05200 |