Finite Semifields and Projective Planes

Author: Knuth, Donald Ervin

Year: 1963

Degree: Dissertation (Ph.D.)

Advisor: Hall, Marshall

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/T3Q6-JC64

Abstract

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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.

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