Smallest Irreducible of the Form $x^2-d{y}^2$

Abstract

It is a classical result that prime numbers of the form $x^2+n{y}^2$ 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 $x^2-d{y}^2$, where $x$, $y$, and $d$ are elements of a polynomial ring over a finite field.