论文标题
在涉及团结根及其应用的多项式上
On a polynomial involving roots of unity and its applications
论文作者
论文摘要
令$ p> 3 $为素数。高斯首先引入了多项式$ s_p（x）= \ prod_ {c}（x-ζ_p^c），$ $ 0 <c <p $和$ c $在所有二次残基$ p $和$ζ_P= e^e^{2πi/p} $方面都会变化。后来，Dirichlet研究了该多项式，并将其用于解决涉及Pell方程的问题。最近，Z.-W Sun研究了一些涉及该多项式的三角身份。在本文中，我们概括了他们的结果。作为我们结果的应用，我们将S. chowla的结果扩展到有关$ \ Mathbb {q}基本单位（\ sqrt {p}）$的一致性，并给出了延长的Ankeny-Arkeny-Artin-Chowla猜想的等效形式。
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.
