Primes of the form x^2+n*y^2 in function fields
Abstract
Let n be a squarefree 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 sgnnormalized rankone 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2009
 arXiv:
 arXiv:0911.2833
 Bibcode:
 2009arXiv0911.2833M
 Keywords:

 Mathematics  Number Theory;
 11D85;
 11E12
 EPrint:
 The paper has been withdrawn by the author