7 Groups, Rings, and Fields

Abstract (or modern) algebra studies basic mathematical structures that appear in many settings . In terms of groups, we study mathematical sets and a single operation (such as addition or multiplication). The study of how two operations interact is contained in studying rings and fields. In this chapter we intersperse the axioms and definitions of abstract algebra with the concrete examples that appear in the K-12 curriculum. The goal of the study of these basic groups, rings, and fields is to better understand the connections between different content in the K-12 curriculum with the goal of improving mathematics teaching by helping students learn new material by making connections with material they have previously learned.