Type of Document 
Master's Thesis 
Author 
Brunson, Jason Cornelius

Author's Email Address 
jabrunso@math.vt.edu 
URN 
etd05102005144432 
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 
20050428 
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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