Date of Award

May 2020

Degree Type

Thesis

Degree Name

Master of Science

Department

Mathematics

First Advisor

Allen Bell

Committee Members

Jeb Willenbring, Yi Ming Zou

Keywords

Dirichlet's Unit Theorem, Number fields, Unit group

Abstract

Let $K$ be a number field of degree $n$. An element $\alpha \in K$ is called integral, if the minimal polynomial of $\alpha$ has integer coefficients. The set of all integral elements of $K$ is denoted by $\mathcal{O}_K$. We will prove several properties of this set, e.g. that $\mathcal{O}_K$ is a ring and that it has an integral basis. By using a fundamental theorem from algebraic number theory, Dirichlet's Unit Theorem, we can study the unit group $\mathcal{O}_K^\times$, defined as the set of all invertible elements of $\mathcal{O}_K$. We will prove Dirichlet's Unit Theorem and look at unit groups for the special case of cubic number fields of type $(1,1)$. The structure of the unit group allows us to define a fundamental unit for this type of field. We will study the relation between the discriminant of the number field and this fundamental unit.

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