Knuth, Donald Ervin (1963) Finite semifields and projective planes. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06042004-141331
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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.))|
|Degree Grantor:||California Institute of Technology|
|Division:||Physics, Mathematics and Astronomy|
|Thesis Availability:||Public (worldwide access)|
|Defense Date:||1 January 1963|
|Default Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Imported from ETD-db|
|Deposited On:||07 Jun 2004|
|Last Modified:||26 Dec 2012 02:51|
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