# 2: Fields and Rings

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You have been exploring numbers and the patterns they hide within them since your earliest school days. In Chapter 1 we reminded ourselves about some of those patterns (with the goal of understanding factorization) and worked to express them in a more formal way. You may find yourself wondering why we are going out of our way to complicate ideas you have understood since elementary school. The reason for the abstraction (and the reason for this course!) is so that we can explore just how far we can push these patterns. How far does our understanding of factorization in the integers stretch to other types of numbers and other mathematical objects (like polynomials)? In this chapter we will set the ground work for answering that question by introducing ideas that will assist us in streamlining our investigation into factorization.

• 2.1: Fields
We now begin the process of abstraction. We will do this in stages, beginning with the concept of a field. First, we need to formally define some familiar sets of numbers.
• 2.2: Rings
In the previous section, we observed that many familiar number systems are fields but that some are not. As we will see, these non-fields are often more structurally interesting, at least from the perspective of factorization; thus, in this section, we explore them in more detail. Before we proceed with that endeavor we will give a formal definition of polynomial so that we can include it in our work.
• 2.3: Divisibility in Integral Domains
When we introduced the notion of integral domain, we said that part of the reason for the definition was to capture some of the most essential properties of the integers.  This is the heart of abstraction and generalization in mathematics: to distill the important properties of our objects of interest and explore the consequences of those properties. One such important property of Z is cancellation.
• 2.4: Principal Ideals and Euclidean Domains
In this section, we begin a set-theoretic structural exploration of the notion of ring by considering a particularly important class of subring which will be integral to our understanding of factorization.

This page titled 2: Fields and Rings is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael Janssen & Melissa Lindsey via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.