## Graduation Semester and Year

2012

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Minerva Cordero-Epperson

## Abstract

The study of semifields was originated by L.E. Dickson in 1905 and was greatly developed in the 1960s by Albert, Knuth, Walker, Kleinfeld among others. Two problems of interest in this area are the primitivity of semifields and the dimension of semifields.For the first problem, we provide an equivalent condition for the right (or left) primitivity of finite semifields and with this result, we find that the classical Knuth binary semifields of order 2^{15}, 2^{17} and 2^{19} are all right and left primitive; the Albert semifields of order 2^i, where i=7, 9, 11, 13 are all primitive. Also we prove that the number of primitive elements in the classical Knuth binary semifields of order 2t, where t is any odd integer between 5 and 19, is a multiple of t.For the second problem, we first prove that the classical Knuth binary semifields can not be fractional dimensional. Then we consider two special classes of the generalized Knuth binary semifields of order 2t, t odd. One class contains GF(22), when t is between 5 and 31, which is fractional dimensional; the other class is commutative and does not contain the subfields GF(22) or GF(23). Finally we show that the Albert semifields An(S) do not contain the subfield GF(23) when (n, 3)=1.

## Disciplines

Mathematics | Physical Sciences and Mathematics

## License

This work is licensed under a Creative Commons Attribution-NonCommercial-Share Alike 4.0 International License.

## Recommended Citation

Chen, Linlin, "On Primitivity And Dimension Of Finite Semifields And Their Planes" (2012). *Mathematics Dissertations*. 40.

https://mavmatrix.uta.edu/math_dissertations/40

## Comments

Degree granted by The University of Texas at Arlington