We will spend the first few weeks of the course examining elementary number theory using the tools and viewpoint of abstract algebra. Such a perspective provides beautiful proofs of many simple facts together with a deeper understanding of fundamental results. From here, we will begin to think about the analog of the integers in fields extending the rationals. Our first step will be to analyze the Gaussian Integers and use their elegant structure to prove nontrivial theorems about the ordinary integers. Following up on that, we will study quadratic/cyclotomic extensions and examine the generalization of the integers in these settings. Unfortunately, we will discover that unique factorization and other essential properties can fail in these higher analogues. Toward the end of the course we will work to understand how to rectify the situation using more advanced ring theory.