标题： 最小不可约形式的$x^2-dy^2$ 显示英文标题

标题： Smallest Irreducible of the Form $x^2-dy^2$

摘要： 一个经典结果是，形如$x^2+ny^2$的素数可以通过类域论来刻画无限集的$n$。 本文推导了这一经典结果的函数域类比。 然后我们应用 Chebotarev 密度定理的有效版本来估计形如$x^2-dy^2$的最小不可约元的次数，其中$x$、$y$和$d$是有限域上多项式环中的元素。 显示英文摘要

摘要： It is a classical result that prime numbers of the form $x^2+ny^2$ can be characterized via class field theory for an infinite set of $n$. In this paper we derive the function field analogue of the classical result. Then we apply an effective version of the Chebotarev density theorem to bound the degree of the smallest irreducible of the form $x^2-dy^2$, where $x$, $y$, and $d$ are elements of a polynomial ring over a finite field.