Definition of a field

We have seen that a group is a set which has a structure admitting the addition or subtraction of any two elements, or alternatively their multiplication and division, but not all four rules. If we use the addition notation and introduce a product we obtain a ring, which still will not necessarily admit of division. If now we extend our requirements to include division, so that all four rules are present, we obtain a structure called a field.

A field then is a ring with additional properties. For division to be possible we must have a unity and also an inverse to each element (except 0). Commutativity of addition is essential for the Distributive Law to be useful (as is the case for general rings) but commutativity of multiplication need not be insisted upon. We find however that nearly all fields of any practical interest possess this property and that if we assume it some of our theory is made simpler, so we will postulate it as one of our field axioms. (Nearly all writers follow the same convention, but not all, and the reader must be careful to check this when reading any new book or paper.) A structure that obeys all the field axioms except for commutativity of multiplication will be called a skew field (sometimes also called a division ring, or even a sfield) and in §4.9 we will indicate ways in which these differ from fields.

Introduction and examples

A very common procedure in abstract algebra is that of obtaining new structures from ones which are already known. We have used this idea many times in the previous chapters and in volume 1—see the examples below.

There are two main purposes in investigating such extensions. The first is so that we can study a structure in terms of its components, which are usually simpler. Thus we are interested in the substructures of a given group, ring or field and in the relationship that the whole structure bears to them. We have used this idea in expressing sets as the direct products of two or more of their subsets.

The second purpose is to obtain new structures, by applying certain more or less standard methods to sets that we already know. We may thus obtain structures that possess some required additional property: for example the integers do not admit of division and we extend these to the field of rationals so as to obtain a set that does so, while still containing a subset isomorphic to the integers. In many cases we use the extension methods in order to obtain rigorously definitions of sets that we already know about intuitively: the complex numbers give a good example, where the rigorous development by number pairs is somewhat tedious but gives us a secure foundation for working with complex numbers, defining them in terms of real numbers, which are already known.

Introduction

In the preceding chapters we have covered most of the elementary theory of abstract algebra, and have indeed on occasions included some fairly deep and difficult properties (for example in the later parts of chapter 4 and in chapter 8). But of course there is an immense amount of work that we have not touched. Much of this is very difficult. We cannot hope to explain in a short chapter even what it is about, but there is some which, while it is not possible (nor desirable in a book at this level) for us to develop it rigorously and fully, is nevertheless amenable to brief indications of its scope and importance. In this chapter we will look at some such topics. Proofs will be omitted, and we will not even give references, since they may be found in the standard works on the various types of structure. There is of course no attempt at completeness in any sense—we merely select various ways in which the work we have done develops and try to show the reader some more of the power and beauty of the methods of the subject.

Most of the work continues the theory of groups, rings and fields.

The isomorphism theorems

When studying the structure of a group it is necessary to consider its subgroups, particularly its invariant subgroups (so that we can form the quotient group, which we expect to be a simpler group than the original).

Description of the problem

The advanced theory of groups depends on the idea of an invariant (or normal or self-conjugate) subgroup. Beyond the elementary work on subgroups leading up to Lagrange's theorem, and homomorphisms, it is not possible to progress far with the study of group theory without introducing and using this concept. Its importance cannot be over-emphasised.

Although so important, the idea of an invariant subgroup is not easy to grasp for a student meeting it for the first time. It is easy to give a definition, but the significance of this and its far-reaching implications are difficult to appreciate before more experience is gained. There are in fact several approaches to the problem.

In this chapter we will develop the essential theory of invariant subgroups. In §6.2 and §6.3 we give what seems to be the most illuminating definition, and in §6.4 give an alternative and important approach. In §6.7 and §6.8 the important connection between invariant subgroups and homomorphisms is explained; in §6.9 the concept is linked with the idea of direct products.

In the remainder of the present section we try to explain some of the ideas that lie behind the definition, attempting to show intuitively some of the importance of the concept. The approach is informal, the rigorous treatment being left until the next sections.

The problem is basically that of ‘dividing’ one group by another: that is in ‘dividing’ a group G by a subgroup H to obtain another group, a ‘quotient group’.

Mappings

This section is a brief résumé of part of chapter 9 of volume 1, and the reader who has that volume to hand should re-read the chapter in question. On the other hand, the inclusion of the material again here makes the succeeding work on homomorphisms self-contained.

A mapping is an extension of the idea of a function to general sets. The function y = f(x) associates a number y to each number x, where the ‘numbers’ x and y are usually real but may be complex, or indeed restricted to the integers or other subsets of the complex numbers. To generalise this idea, a mapping of the set A into the set B associates a unique element b (in B) to each element a in A. The sets A and B may be the same or different. For a mapping to be properly defined we must have a unique b associated with each element of A, The elements b may be all different or not, so that two or more elements of A may give rise to the same element of B. But we must not have two b's corresponding to the same a.

Notation

A is called the object space of the mapping, B is the image space, a is the object whose image is b, and we say that we have a mapping of A into B.

Introduction—the algebra of sets and the algebra of statements

In 1854 George Boole published The Laws of Thought, in which he showed that many of the basic ideas of logic could be expressed in symbolical terms and that laws similar to the ordinary rules of algebra could be applied to these symbols. This book was one of the most important and far reaching in the history of mathematics: for the first time it was seen that algebra, hitherto applied almost exclusively to numbers or similar mathematical entities, was in fact of much wider application. Abstract algebra could be said to have begun with Boole.

The algebra that Boole used to express logical ideas, when formalised into an abstract system in the modern manner, is now known as Boolean algebra. Besides having local applications, it is the natural algebra for work in the theory of sets, and other applications have also been discovered.

In this chapter we define in §11.2 a Boolean algebra as an abstract system and consider some simple consequences of the laws in the following section. In § 11.4 we investigate the idea of a Boolean function, continuing this study in §11.5. In §11.6 we consider some applications to logic, and in §11.7 indicate that any finite Boolean algebra may be represented as an algebra of sets. In the final sections we investigate briefly some other applications and a related idea, that of a lattice.

Introduction—equivalence relations

In this first chapter we repeat briefly some of the basic work on group theory that was done in detail in volume 1. The reader who wishes to study the elementary theory in detail should read volume 1 and omit this chapter, but the student who possesses a fair amount of mathematical maturity and who wants to study other types of algebraic structure but who needs first to know the standard theorems and definitions of group theory may substitute this chapter, which will provide him with the knowledge he requires, though not with all the detailed examples and explanations that were given in the first volume.

We assume that the reader is familiar with set notation. We use the standard notation, but the following points should be noted.

Inclusion. We denote the fact that A is contained in B by A ⊆ B, reserving A ⊂ B to mean A ⊆ B but A ≠ B.

The empty and the universal sets. The empty set is denoted by ø, and we use no special notation for the universal set.

Complement and difference. The complement of A is written as A′; the difference (A − B) means the set of elements in the universal set that are in A but not in B: it does not imply that B ⊆ A.

Sets in terms of their elements. The set containing elements a, b, c, …, k is denoted by {a, b, c, …, k}.

Euclidean geometry

There are several ways of placing the various types of geometry on a rigorous axiomatic footing. One way is by means of purely geometrical axioms: thus projective geometry may be based on axioms of incidence, together with one or two others, such as that postulating the idea of cross-ratio. One very satisfying method is by means of algebra: basing geometry on the idea of a vector space. The connection between a vector space and geometry based on a co-ordinate system is obvious, and since it is easy to define the former abstractly, as in the previous chapter, it is comparatively straightforward to give a rigorous development of a geometry by this means. Such a treatment loses something in beauty—analytical geometry is never as fascinating as pure geometry—but it gains tremendously in the ease with which firm foundations may be laid. In this section we will indicate briefly the formulation of Euclidean geometry by this means, and in the following sections we will apply the method to some other geometries.

Euclidean geometry is a complicated structure, much more so than is, for example, projective geometry. Even if we define lines, planes etc., in terms of linear manifolds in the obvious ways the ideas that we have already introduced about vector spaces are not sufficient for our present purpose. The idea that distinguishes Euclidean geometry, and which we have not so far introduced, is that of distance, and its consequence, the notion of angle.

Definition of a ring

We have seen that in a group we have a set together with an operation which obeys certain rules analogous to the rules of ordinary multiplication, or addition. It is immaterial whether we call this operation addition or multiplication, except that it is conventional to use the product notation in the general case, reserving the sum notation for Abelian groups.

It often happens that we have a set which contains two different operations, and that these operations are linked in some way. The obvious example is in the set of real numbers where we have addition and multiplication, which are linked by the Distributive Law. Subtraction arises from addition (by the introduction of negatives) and division from multiplication (by the introduction of inverses), but the Distributive Law forms the only real connection between sum and product, and distinguishes between them since the dual Distributive Law (that x + yz = (x + y) (x + z), obtained by interchanging sum and product) is not satisfied.

The set of reals is by no means the only set containing two operations. Other examples are given below.

Example 1. The set of integers admits of addition (and subtraction) and multiplication (but not division).

Example 2. The set of polynomials in x (of any degree, with real coefficients) admits of addition and multiplication, including subtraction but not division in general, though it may do so in special cases.

Introduction

The study of vectors, particularly of their algebraic properties, is often best performed by considering the set of all vectors of a given type (for example that of all three-dimensional vectors with real co-ordinates) and the algebraic structure which these comprise. Such a structure is called a vector space and, as we expect, may be defined by taking as axioms a very few fundamental laws which we require it to satisfy. It turns out that, in the ‘finite dimensional’ case (we anticipate a little in the present section the ideas which will be developed formally later in the chapter), our basic axioms characterise completely sets of what we would naturally think of as ‘vectors’: in other words they lead to no structures that are not immediately recognizable as sets of vectors in the elementary sense. (This is quite different from the case in group theory, where there is an extraordinary richness of possible structure, and from fields, where we lay down a large collection of axioms and still obtain a fairly wide diversity of structure.) The advantage of approaching the subject axiomatically is, as always, that we know precisely what we may assume and with what we are dealing: the work follows rigorously from the axioms and we are made aware of the distinction between fundamental properties and derived ones.

We come now to an important point. Any advanced, or even fairly advanced, work on vectors leads inevitably to the study of matrices, and to the whole vast subject of what is known as ‘linear algebra’.

Definition and elementary properties

In the two previous chapters we have discussed rings and fields, the former being structures which admit of addition, subtraction and multiplication, while the latter possess a division process also. We have seen that rings possess comparatively little elementary theory, whereas elementary field theory is of course similar to elementary algebra.

There are some rings, however, which are not fields yet possess many properties that are not shared by all rings. Thus the rings of integers and polynomials over any field possess a great deal of important theory (that concerned with primes and factorisation) that is not applicable to rings in general. There are comparatively few such rings (the integers and polynomials form the main examples) but they are so important that it is worthwhile giving them a distinct name and unifying the theory, rather than dealing with each example separately. (We saw this process at work in volume 1, where much of the polynomial theory in chapter 7 was an exact replica of that for integers in chapter 4.) We are thus led to the idea of an integral domain.

We will see that the theory of factorisation in a ring depends basically on the Cancellation Law of multiplication, that xy = xz ⇒ y = z or x = 0. This is not possessed by all rings (for example the residues modulo a composite number and the rings of functions discussed in chapter 3 do not have the property) but it is a property of the integers and polynomials over a field.

This book is the second of a two volume work which attempts to give a broad introduction to the subject of abstract algebra. It assumes no previous knowledge of this work but does expect a fair amount of mathematical sophistication. Thus it is not intended to be used as a textbook at an elementary level in schools, as part of one of the ‘modern mathematics’ courses, but is aimed rather at a fairly intelligent sixth-former, who has been brought up on either traditional or modern syllabuses and who wishes to know something more about modern algebra, for example as a preparation for university work. This second volume in particular should be found useful by undergraduates, as giving a broad background before a study in more depth is undertaken, and it does in fact cover a great deal of work usually included in university first degree courses, though not all such work. Teachers in schools and students in teacher training colleges who wish to learn some abstract algebra as an aid and background to their teaching should also find parts of both volumes interesting and useful. Numerous books on the subject have been and are being written, but these are usually either too advanced, with a lack of motivation, for the beginner, or are written at a far less sophisticated level, for younger pupils. It is still not easy to find many which start from the beginning and lead up to substantial and important ideas gradually and in a fairly elementary manner.