Date of Award
5-2026
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematical Sciences
Committee Chair/Advisor
James Coykendall
Committee Member
Ryann Cartor
Committee Member
Hui Xue
Committee Member
Robert Dicks
Abstract
This dissertation explores the arithmetic of numerous algebraic objects living within an algebraic number field from submonoids of the integers up to localizations of the ring of integers. We begin with a study of factorization in proper orders using an element-theoretic approach. In Chapter 2, by defining a natural generalization of the Davenport constant, we are able to determine the elasticity of certain orders whose integral closure is a unique factorization domain. In Chapter 3, using ideal-theoretic analogues, we are able to significantly broaden the scope of our results and the literature on factorization in orders. In particular, we give a complete characterization of the elasticity of an any order with prime conductor in terms of its class group. Chapter 4 focuses on the action of the Galois group of a Galois number field on the class group. This action can be used to place significant restrictions on the structure of the latter and has implications for the arithmetic of the ring of integers and its normset.
Recommended Citation
Kettinger, Jared, "Galois Action and Arithmetic in Algebraic Number Fields" (2026). All Dissertations. 4284.
https://open.clemson.edu/all_dissertations/4284
Author ORCID Identifier
0009-0006-4120-6883