Type of Document 
Dissertation 
Author 
Maciak, Piotr

Author's Email Address 
pmaciak@math.lsu.edu 
URN 
etd04272010174411 
Title 
Primes of the Form X^2+nY^2 in Function Fields 
Degree 
Doctor of Philosophy (Ph.D.) 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Morales, Jorge 
Committee Chair 
Litherland, Richard 
Committee Member 
Perlis, Robert 
Committee Member 
Verrill, Helena 
Committee Member 
Wolenski, Peter 
Committee Member 
Rau, A. Ravi P. 
Dean's Representative 

Keywords 
 weak representation
 strong representation
 principal form
 normalizing field
 jinvariant

Date of Defense 
20100416 
Availability 
unrestricted 
Abstract
Let n be a squarefree polynomial over F_q, where q is an odd prime power. In this work, 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.

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