On a polynomial involving roots of unity and its applications
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866910851975348224 |
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| author | Wu, Hai-Liang She, Yue-Feng |
| author_facet | Wu, Hai-Liang She, Yue-Feng |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2001_02860 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On a polynomial involving roots of unity and its applications Wu, Hai-Liang She, Yue-Feng Number Theory 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. |
| title | On a polynomial involving roots of unity and its applications |
| topic | Number Theory |
| url | https://arxiv.org/abs/2001.02860 |