ABSTRACT
In this chapter, the authors consider the more general situation of quadratic fields and algebraic integers. They explain a situation where there is no unique factorization into irreducibles and show how this causes difficulties. The author deal with the beginning of the subject called algebraic number theory, which started in the 1800s and is an active area of research. The integers are a subset of the rational numbers and the authors can add, subtract, and multiply integers and obtain integers as answers. In order to work with situations where there is not unique factorization, Ernst Eduard Kummer and Richard Dedekind in the 1800s developed the theory of ideals, a basic concept in algebraic number theory. This led to abstract ring theory and has had a substantial influence on modern mathematics.
