## Graduation Semester and Year

2014

## Language

English

## Document Type

Dissertation

## Degree Name

Doctor of Philosophy in Mathematics

## Department

Mathematics

## First Advisor

Minerva Cordero

## Abstract

It is well known that any finite semifield, S, can be viewed as an n-dimensional vector space over a finite field or prime order, Fp, and that the multiplication in S defines and can be defined by an n x n x n cubical array of scalars, A. For any element a E S, the matrix, La, corresponding to left multiplication by a can be determined from A. In this paper we show that there exists a unique monic polynomial of minimal degree, f E Fp[x], such that f(a) = 0, and which divides the minimal polynomial of La. Furthermore, we show that some properties of f in Fp[x] correspond to properties of a in S. These results, in turn, help optimize a method we introduce which uses A to determine the automorphism group of S. We show that under certain conditions A can be inflated to define a new semifield, S[m], over the field Fpm , and that inflation preserves isotopism and isomorphism between inflated semifields. Finally, we apply our results to the 16-element semifields, and give algebraic constructions for each of these semifields for which no construction currently exists.

## 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

Aman, Kelly Casimir, "Applications Of Cubical Arrays In The Study Of Finite Semifields" (2014). *Mathematics Dissertations*. 140.

https://mavmatrix.uta.edu/math_dissertations/140

## Comments

Degree granted by The University of Texas at Arlington