The Rational Numbers

Abstract

The operations addition and multiplication on the domain of integers lead to the algebraic concept of a ring. The study of the fundamental aspects of ring theory is the subject of this chapter. We study rings, subrings, ideals, ring homomorphisms, and quotient rings, and discover special classes of rings known as integral domains and fields, which again play an important role in the construction of number systems. We show that every integral domain can be enlarged essentially uniquely to a field. Since the ring of integers turns out to be an integral domain, we are thus able to enlarge it to the field of rational numbers. The appendix to this chapter studies the problem of the existence of rational solutions to polynomial equations leading to classical questions in arithmetic geometry, which in part were resolved only recently and some of which are topics of current research.