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Finite Semifields and Projective Planes


Knuth, Donald Ervin (1963) Finite Semifields and Projective Planes. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/T3Q6-JC64.


NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This paper makes contributions to the structure theory of finite semifields, i.e., of finite nonassociative division algebras with unit. It is shown that a semifield may be conveniently represented as a 3-dimensional array of numbers, and that matrix multiplications applied to each of the three dimensions correspond to the concept of isotopy. The six permutations of three coordinates yield a new way to obtain projective planes from a given plane. Several new classes of semifields are constructed; in particular one class, called the binary semifields, provides an affirmative answer to the conjecture that there exist non-Desarguesian projective planes of all orders 2[...], if n is greater than 3. With the advent of binary semifields, the gap between necessary and sufficient conditions on the possible orders of semifields has disappeared.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Turing Award, 1974. National Medal of Science, 1979. Harvey Prize, 1995. Kyoto Prize in Advanced Technology, 1996.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Hall, Marshall
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1963
Record Number:CaltechETD:etd-06042004-141331
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2441
Deposited By: Imported from ETD-db
Deposited On:07 Jun 2004
Last Modified:04 Jan 2024 18:58

Thesis Files

PDF (Knuth_de_1963.pdf) - Final Version
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