Presentation Title

Harmony Amid Chaos

Faculty Mentor(s)

Dr. Justin Brown

Project Type

Other

Scholarship Domain(s)

Scholarship of Discovery

Presentation Type

Presentation

Abstract

Presentation Location: Weber Center, Room 101

Abstract

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 ��2 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 well as (what we believe is) an improvement of his method, the second by statistical analysis of the Galois Fields at �� = 17, the highest prime value we were able to generate Galois Fields of size ��2 for.

Permission type

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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Apr 16th, 3:25 PM Apr 16th, 3:55 PM

Harmony Amid Chaos

Other

Presentation Location: Weber Center, Room 101

Abstract

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 ��2 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 well as (what we believe is) an improvement of his method, the second by statistical analysis of the Galois Fields at �� = 17, the highest prime value we were able to generate Galois Fields of size ��2 for.