Date of Award

2026

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Tamar Friedmann

Abstract

This thesis derives a generating function that describes the matrices of a given multiplicative order over finite fields of a given order assuming that the order of a field is not a divisor of the desired order of the matrix. This is done by using the power series known as the cycle index derived from the rational canonical form. This index can be factored, and using the fact that the minimal polynomial divides some polynomial of the form x^k − 1 we can show that the minimal polynomial is squarefree. This sufficiently restricts the form of the rational canonical form to allow us to enumerate the matrices of a given multiplicative order k given the factorization of x^k −1

Keywords

matrix, multiplicative order, finite field, square-free factorization, rational canonical form, generating functions

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