Publication Date

Fall 2018

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

Advisor

Jordan Schettler

Subject Areas

Mathematics

Abstract

Let $K$ be a number field and let $\OO_K$ denote its ring of integers. We can define a graph whose vertices are the elements of $\OO_K$ such that an edge exists between two algebraic integers if their difference is in the units $\OO_K^{\times}$. Lenstra showed that the existence of a sufficiently large clique (complete subgraph) will imply that the ring $\OO_K$ is Euclidean with respect to the field norm. A recent generalization of this work tells us that if we draw more edges in the graph, then a sufficiently large clique will imply the weaker (but still very interesting) conclusion that $K$ has class number one.

This thesis aims to understand this new result and produce further examples of cliques in rings of integers. Lenstra, Long, and Thistlethwaite analyzed cliques and gave us class number one through a prime element. We were able to extend and generalize their result to larger cliques through prime power elements while still preserving our desired property of class number one. Our generalization gave us that class number one is preserved if the number field $K$ contained a clique that is generated by a prime power.

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