Introduction to Division Rings

Abstract

In the category of rings, the most “perfect” objects are the rings in which we can not only add, subtract, and multiply, but also divide (by nonzero elements). These rings are called division rings, or skew fields, or sfields. No matter how we call them, it is clear that a careful study of their properties would be vital for the development of ring theory in general. In this introductory chapter, we shall give an exposition on the basic theory of division rings, starting with Wedderburn’s beautiful theorem that any finite division ring is commutative. This landmark result, proved by Wedderburn in 1905, has fascinated generations of algebraists and inspired a long sequence of more general commutativity theorems, by Jacobson, Kaplansky, Herstein, and others. We shall study some of these results in §13, and go on to study maximal subfields in division rings, polynomial equations over division rings, and ordered division rings in §§15, 16, and 17. In §14, we present several types of elementary constructions of division rings, thus providing some basic examples with which to illustrate the general theory. This section is written independently of §13, so it is possible for the reader to start this chapter by first reading §14 to see the basic examples before reading §13.