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

TL;DR

This paper extends classical number theory results to function fields, providing bounds on the smallest irreducible of the form $x^2-dy^2$ over polynomial rings in finite fields using effective Chebotarev density theorem.

Contribution

It derives the function field analogue of classical results on primes of the form $x^2+ny^2$ and applies an effective Chebotarev density theorem to bound the degree of the smallest such irreducible.

Findings

Bound on the degree of the smallest irreducible of the form $x^2-dy^2$

Function field analogue of classical prime characterization

Application of effective Chebotarev density theorem

Abstract

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.