Primes of the form x^2+n*y^2 in function fields

TL;DR

This paper characterizes which irreducible polynomials over finite fields can be expressed as x^2 + n y^2, using class field theory and Drinfeld modules, extending classical number theory results to function fields.

Contribution

It provides explicit criteria and an algorithm for representing polynomials as x^2 + n y^2 in function fields, generalizing classical results and previous algorithms.

Findings

Representation governed by splitting in Hilbert class field

Necessary condition involves splitting of p in H

Algorithm for constructing generating polynomial for H/K

Abstract

Let n be a square-free polynomial over F_q, where q is an odd prime power. In this paper, we determine which irreducible polynomials p in F_q[x] can be represented in the form X^2+nY^2 with X, Y in F_q[x]. We restrict ourselves to the case where X^2+nY^2 is anisotropic at infinity. As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary (and almost sufficient) condition is that the ideal generated by p splits completely in the Hilbert class field H of K = F_q(x,sqrt{-n}) (for the appropriate notion of Hilbert class field in this context). In order to get explicit conditions for p to be of the form X^2+nY^2, we use the theory of sgn-normalized rank-one Drinfeld modules. We present an algorithm to construct a generating polynomial for H/K. This algorithm generalizes to all situations an algorithm of D.S. Dummit and D.Hayes for the case where -n is monic of odd degree.