Number Theory Commons^™

Harmony Amid Chaos, Drew Schaffner

Pence-Boyce STEM Student Scholarship

We provide a brief but intuitive study on the subjects from which Galois Fields have emerged and split our study up into two categories: harmony and chaos. Specifically, we study finite fields with elements where is prime. Such a finite field can be defined through a logarithm table. The Harmony Section is where we provide three proofs about the overall symmetry and structure of the Galois Field as well as several observations about the order within a given table. In the Chaos Section we make two attempts to analyze the tables, the first by methods used by Vladimir Arnold as …

Solutions To Systems Of Equations Over Finite Fields, Rachel Petrik

Theses and Dissertations--Mathematics

This dissertation investigates the existence of solutions to equations over finite fields with an emphasis on diagonal equations. In particular:

1. Given a system of equations, how many solutions are there?
2. In the case of a system of diagonal forms, when does a nontrivial solution exist?

Many results are known that address (1) and (2), such as the classical Chevalley--Warning theorems. With respect to (1), we have improved a recent result of D.R. Heath--Brown, which provides a lower bound on the total number of solutions to a system of polynomials equations. Furthermore, we have demonstrated that several of our lower bounds …