| 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 |
|
| Keywords |
- weak representation
- strong representation
- principal form
- normalizing field
- j-invariant
|
| Date of Defense |
2010-04-16 |
| Availability |
unrestricted |
Abstract
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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| Filename |
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Approximate Download Time
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56K Modem |
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maciak_diss.pdf |
340.95 Kb |
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