#### Date of Award

May 2020

#### Document Type

Thesis

#### Degree Name

Master of Science (MS)

#### Department

Mathematical Sciences

#### Committee Member

James Coykendall

#### Committee Member

Sean Sather-Wagstaff

#### Committee Member

Matthew Macauley

#### Abstract

We consider integral domains $R\subset S \subset T$, where $T$ is integral over $R$. In particular, we study the behavior of an element $p \in R$ which is prime in both $R$ and $T$ and which may or may not be prime in $S$. If $p$ remains prime in $S$, we say that $p$ has prime stability. Otherwise, $p$ does not have prime stability. We first consider the case of a quadratic ring of integers, $\ints \subset \ints[\omega]$, with $\ints[\omega]$ a quadratic extension of $\ints$. We proceed to prove results using Legendre, Jacobi, and Kronecker symbols and field discriminants, using both of these tools to determine a criterion for determining the primality of an element in orders of the ring of integers, namely the intermediate extension in the extension $\ints \subset \ints[n\omega] \subset \ints[\omega]$. After this, we consider some conjectures concerning the behavior of prime stability in integral extensions. We then discuss some fundamental definitions and previous work on integral extensions. With all of these tools, we observe some examples which provide support for the conjectures.

#### Recommended Citation

de Castro, Philip, "Prime Stability in Intermediate Extensions of Integral Domains" (2020). *All Theses*. 3343.

https://tigerprints.clemson.edu/all_theses/3343