Title page for ETD etd-04272010-174411

Type of Document Dissertation
Author Maciak, Piotr
Author's Email Address pmaciak@math.lsu.edu
URN etd-04272010-174411
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
  • weak representation
  • strong representation
  • principal form
  • normalizing field
  • j-invariant
Date of Defense 2010-04-16
Availability unrestricted
Let n be a square-free 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 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.
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