Groups, Rings and Fields

The notions of a ‘set’ and of a ‘mapping’ are fundamental in modern mathematics. In many mathematical contexts a perceptive choice of appropriate sets and mappings may lead to a better understanding of the underlying mathematical processes. We shall outline those aspects of sets and mappings which are relevant to present purposes and, for the delectation of the reader, conclude with a few logical paradoxes in regard to sets.

Numbers are encountered early in life and in many practical contexts. In our infancy we develop a feeling for the natural numbers by chanting “one, two, three, …”, “eins, zwei, drei, …”, “yî, èr, sân,…”, or their equivalent, in whatever may be our mother tongue. By childhood we have assimilated, without too much conscious effort, the elementary properties of the addition, subtraction, multiplication and division of the natural numbers; in this text we take these elementary properties for granted. In such an approach we are following the celebrated dictum of L. Kronecker (1823–91), namely, “Die ganze Zahl schuf der liebe Gott; alles übrige ist Menschenwerk” which we may render in English as “God created the integers; everything else is man’s handiwork”. (Quotation from Philosophie der Mathematik und Naturwissenschaft by H. Weyl, R. Oldenburg, München, 1928, Section 6, page 27).

In the previous chapter we have seen that the integers possess a division algorithm and that from this division algorithm there may be derived a Euclidean Algorithm for finding the greatest common divisor of two given integers. ‘Polynomials’ share many properties in common with the integers, having a division algorithm and a corresponding Euclidean Algorithm. As our treatment of polynomials proceeds, initially somewhat informally, it will become apparent that we need to consider much more precisely the extent to which integers and polynomials share common features. In this way we shall be led to enunciate axioms for an algebraic system called a ‘ring’ and for a ring of a particular type called an ‘integral domain’ which incorporates some of the features common to integers and polynomials.

In Chapter 3, we began to consider an axiomatic treatment of certain algebraic concepts and operations; in particular in the case of a ring we needed to consider two binary operations. Since simplicity usually has some advantages we shall in this chapter consider only one binary operation. We begin therefore with the notion of a ‘semigroup’ from which we shall be led to consider a ‘group’, one of the most fundamental structures in modern mathematics.

We first formulated the abstract concept of a ring from our considerations of the integers and of polynomials. It is our aim to carry forward and to develop the concept of a ring, for which the integers will continue to serve as a useful exemplar.

In this final chapter we extend our knowledge of group theory. Among other aspects of finite groups, we investigate permutation groups and obtain two results of cardinal importance in the theory of finite groups. In the first of these, we establish the structure of Abelian groups and, in the second, we establish the existence of the so-called ‘Sylow p-subgroups’ of a finite group.