Carlitz Cyclotomic Theory over Rational Function Fields

Open Access
Author:
Van Hook, Jacob Arthur
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Mihran Papikian, Thesis Supervisor
  • Sergei Tabachnikov, Honors Advisor
Keywords:
  • Carlitz
  • cyclotomic
  • function field
Abstract:
This thesis explores properties of the traditional cyclotomic polynomials in the setting of the rational numbers such as their irreducibility and the integral nature of their coefficients. These properties are understood and then translated to a similar theory about polynomials over a finite field and extensions of their fraction field. This is related to a larger relationship between the integers and polynomials over a finite field. These translated results are from Rosen’s Number Theory in Function Fields, and I have restated these results and supplemented them with context from other sources for their presentation here.