Title page for ETD etd-05102005-144432


Type of Document Master's Thesis
Author Brunson, Jason Cornelius
Author's Email Address jabrunso@math.vt.edu
URN etd-05102005-144432
Title On Projective Planes & Rational Identities
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Farkas, Daniel R. Committee Chair
Brown, Ezra B. Committee Member
Shimozono, Mark M. Committee Member
Keywords
  • rational identity
  • intersection theorem
  • projective plane
Date of Defense 2005-04-28
Availability unrestricted
Abstract
One of the marvelous phenomena of coordinate geometry is the equivalence of Desargues' Theorem to the presence of an underlying division ring in a projective plane. Supplementing this correspondence is the general theory of intersection theorems, which, restricted to desarguian projective planes P, corresponds precisely to the theory of integral rational identities, restricted to division rings D. The first chapter of this paper introduces projective planes, develops the concept of an intersection theorem, and expounds upon the Theorem of Desargues; the discussion culminates with a proof of the desarguian phenomenon in the second chapter. The third chapter characterizes the automorphisms of P and introduces the theory of polynomial identities; the fourth chapter expands this discussion to rational identities and cements the ``dictionary''. The last section describes a measure of complexity for these intersection theorems, and the paper concludes with a curious spawn of the correspondence.
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