Title page for ETD etd-05182000-12080004

Type of Document Master's Thesis
Author Landquist, Eric Jon
Author's Email Address elandqui@vt.edu
URN etd-05182000-12080004
Title On Nonassociative Division Rings and Projective Planes
Degree Master of Science
Department Mathematics
Advisory Committee
Advisor Name Title
Farkas, Daniel R. Committee Chair
Brown, Ezra A. Committee Member
Green, Edward L. Committee Member
  • division rings
  • semifields
  • projective planes
  • nonassociative
Date of Defense 2000-05-18
Availability unrestricted
An interesting thing happens when one begins with the axioms of a field, but does not require the associative and commutative properties. The resulting nonassociative division ring is referred to as a ``semifield" in this paper. Semifields have intimate ties to finite projective planes. In short, a finite projective plane with certain restrictions gives rise to a semifield, and, in turn, a finite semifield can be used via a coordinate construction, to build a special finite projective plane. It is also shown that two finite semifields provide a coordinate system for isomorphic projective planes if and only if the semifields are isotopic, where isotopy is a relationship similar to but weaker than isomorphism.

Before we prove those results, we explore the nature of isotopy to get a little better feel for the concept. For example, we look at isotopy for associative algebras. We will also examine a particular family of semifields and gather concrete information about solutions to linear equations and isomorphisms.

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