Daniel Packer

Date of Award


First Advisor

William Dunbar

Second Advisor

Timothy Susse


This thesis examines the relationship between elements of integral domains and polynomials over those domains. Alternatively, we can view the same problem as examining the conditions necessary for a specific element to be representable as a sum of specific factors of powers of another element. The first chapter provides a summary of topics to be covered without proof or much examination of the precise structure. The second chapter describes the Algebraic and Number Theoretical bases that the remainder of the thesis is built around. Next, the third chapter introduces well-researched properties of prime-generating polynomials. The final chapter presents original research by the author that builds upon the findings presented in the second and third chapters.