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

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

2009-11-15

arxiv.org/abs/0911.2833v2

Mathematics

Number Theory

The paper has been withdrawn by the author

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Primes of the form x^2+n*y^2 in function fields does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Primes of the form x^2+n*y^2 in function fields, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Primes of the form x^2+n*y^2 in function fields will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-323604