Basic Algebra

Much of algebra can be done using only very little set theory; all that is needed is a means of comparing infinite sets, and the axiom of choice in the form of Zorn’s lemma. These topi cs occupy Sections 1.1 and 1.2. They are followed in Section 1.3 by an introduction to graph theory. This is an extensive theory with many applications in algebra and elsewhere; all we shall do here is to present a few basic results, some of which will be used later, which convey the flavour of the topic.

Our readers will have met groups before, so we shall be fairly brief in recalling the fundamentals, which occupy Sections 2.1-2.3. The remainder of this chapter deals with some notions of importance in elucidating the structure of groups, such as solubility, nilpotence (Section 2.4) and commutator subgroups (Section 2.5). In Section 2.6 we describe the constructions of Frattini and Fitting, which have their counterpart in rings in the form of the radical.

The subsets of a set permit operations quite similar to those performed on numbers. If for the moment we denote the union of two subsets A, B by A +B and their intersection by AB, a notation that will not be used later (despite some historical precedents), then we have laws like AB = BA, A(B + C) = AB + AC, similar to the familiar laws of arithmetic, as well as new laws such as A + A = A, A + BC = (A + B)(A + C). The algebra formed in this way is called a Boolean algebra, after George Boole who introduced it around the middle of the 19th century, and who made the interesting observation that Boolean algebras could also be used to describe the propositions of logic.

Linear algebra deals with fields and vector spaces; here we are concerned with the generalizations to rings and modules over them. Whereas a vector space over a given field is determined up to isomorphism by its dimension, there is much greater variety for modules. Another way of regarding modules is as abelian groups, written additively, with operators. This means that much of general group theory applies, and after recalling the isomorphism theorems , proved for groups in Section 2.3, we treat a number of special situations. Semisimple modules (Section 4.3) come closest to vector spaces; they are direct sums of simple modules, but we have to bear in mind that over a given ring there may be more than one type of simple module. In the free modules (Section 4.6) we have another generalization of vector spaces. The homological treatment of module theory requires the notions of projective and injective module (Section 4.7), and they can usefully be introduced here, as they will occur again in Chapter 10, although their main use will be in FA. Other important notions introduced here are those of matrix ring (Section 4.4) and tensor product (Section 4.8).

Historically the first rings to be studied (in the second half of the 19th century) were the rings of integers in algebraic number fields. At about the same time the theory of algebras began to develop; its most important landmarks were the Wedderburn structure theorems for semisimple algebras, and the study of the radical. The theories merged when it was realized that the Wedderburn theorems could be stated more generally for Artinian rings. This is the form in which the results will be presented here, in Section 5.2 and 5.3; the formulation for general rings (Jacobson radical and density theorem) will be reserved for FA. As a preparation for this study we examine the form the tensor product takes for algebras in Section 5.4, and in Section 5.5 we introduce scalar invariants. In Section 5.6 algebras are used to define an important number-theoretic function, the Mobius function.

Polynomial rings form a simple example of a graded algebra; such algebras occur frequently and in Section 6.1 we define this concept . Another important example is given by free algebras, which are discussed in Section 6.2, as well as the related notions of tensor algebra and symmetric algebra on a K-module . A graded algebra has an important invariant, its Hilbert series, essentially a power series whose coefficients indicate the dimensions of the components. In Section 6.3 we show that the Hilbert series of a commutative Noetherian ring is a rational function and also prove the Golod-Shafarevich theorem, giving a sufficient condition for a graded algebra to be infinite-dimensional. The applications, to construct a finitely generated algebra which is nil but not nilpotent, and a finitely generated infinite p-group , are sketched in the exercises. Finally Section 6.4 deals with exterior algebras, providing a simple derivation of determinants, and giving a brief geometrical application.

Fields form one of the basic algebraic concepts, for which there is an extensive theory, dealing mainly with the form taken by field extensions. In Section 7.1 field extensions are described, and the special case of splitting fields is introduced in Section 7.2, leading to the notion of algebraic closure (Section 7.3). In Section 7.4 we examine the problems arising in finite characteristic. One of the main tools in this study is Galois theory and this forms the subject of Sections 7.5 and 7.6, while Sections 7.10 and 7.11 bring its application to the solution of equations . The special case of finite fields is studied in Section 7.8, using information on the roots of unity (Section 7.7); Section 7.9 is devoted to generators and some invariants of extensions.

Most readers will have met inner products before. Here we take up the subject in a more general form and look at the properties of quadratic forms over a general field and its group of isometries (Sections 8.1-8.3). With each quadratic form a certain algebra is associated, the Clifford algebra, and with the set of all forms on a field the Witt group is associated; these form the subject of Section 8.4 and Section 8.5 respectively, with a further development, the Witt ring of a field, in Section 8.9. In Section 8.10 we take a brief look at symplectic groups and in Section 8.11 we consider quadratic forms in characteristic 2. We also briefly discuss the related topic of ordered fields in Section 8.6, leading to a construction of the real numbers in Section 8.7 and formally real fields in Section 8.8. Some of the later topics are included for completeness, but do not really have a place in a basic account; thus at a first reading the later parts of Sections 8.7-8.11 can be omitted.

Valuation theory may be described as the study of divisibility (in commutative rings) in its purest form, but that is only one aspect. The general formulation leads to the introduction of topological concepts like completion, which provides a powerful tool. It also emphasizes the parallel with the absolute value on the real and complex numbers. After the initial definitions (in Section 9.1) we shall prove the essential uniqueness of the absolute value on R and C (in Section 9.2) and go on to describe the p-adic numbers in Section 9.3 and integral elements in Section 9.4, before looking at simple cases of the extension problem in Section 9.5.

Commutative ring theory has its origins in number theory and algebraic geometry in the 19th century. Today it is of particular importance in algebraic geometry, and there has been an interesting interaction of algebraic geometry and number theory, using the methods of commutative algebra. Here we can do no more than describe the basic techniques and take the first steps in the subject. In Section 10.1 we define the various operations on ideals and use them in Section 10.2 to study unique factorization . In Section 10.3 we give an account of fractions and examine the effect of chain conditions in Section lOA. Many rings of algebraic numbers fail to have unique factorization of elements, but instead have unique factorization of ideals, and the consequences are studied in Sections 10.5 and 10.6. Sections 10.7–10.10 deal with the properties of rings used in algebraic geometry (but also of importance in commutative ring theory): equations (Section 10.7), decomposition of ideals (Section 10.8), dimension (Section 10.9) and the relation between ideals and algebraic varieties (Section 10.10).

Chapter 7 was almost entirely devoted to field extensions of finite degree and concentrated on Galois theory . However, even an introductory account should make some mention of infinite field extensions, and we shall discuss them in the present chapter, including transcendental extensions (Section 11.3) and infinite Galois theory (Section 11.8). The notion of algebraic dependence has similarities to linear dependence, which are described in abstract form in Section 11.1 and applied in Section 11.2. In addition there are concise accounts of topics that are useful in commutative ring theory and algebraic geometry, besides being of independent interest : separability (Sections 11.4 and n .s), the interactions of two or more subfields (Sections 11.6 and 11.7), applications of Galois theory (Section 11.9) and abelian extensions of finite exponent (Section 11.10).