Introduction

Algebra is essentially concerned with calculating, that is, performing, on elements of a set, "algebraic operations", the most well-known example of which is provided by the "four rules" of elementary arithmetic.

This is not the place to retrace the slow process of progressive abstraction by which the notion of algebraic operation, at first restricted to natural numbers and measurable magnitudes, little by little enlarged its domain, whilst the notion of "number" received a parallel generalization, until, passing beyond the latter, it came to be applied to elements with no "numerical" character, for example permutations of a set (see the Historical Note to Chapter I). It is no doubt the possibility of these successive extensions, in which the form of the calculations remained the same, whereas the nature of the mathematical entities subjected to these calculations varied considerably, which was responsible for the gradual isolation of the guiding principle of modern mathematics, namely that mathematical entities in themselves are of little importance; what matters are their relations (see Book I). It is certain in any case that Algebra attained this level of abstraction well before the other branches of Mathematics and it has long been accustomed to being considered as the study of algebraic operations, independent of the mathematical entities to which they can be applied.

Deprived of any specific character, the common notion underlying the usual algebraic operations is very simple: performing an algebraic operation on two elements $a, b$ of the same set E, means associating with the ordered pair $(a, b)$ a well-defined third element $c$ of the set E. In other words, there is nothing more in this notion than that of function: to be given an algebraic operation is to be given a function defined on $E \times E$ and taking its values in E; the only peculiarity lies in the fact that the set of definition of the function is the product of two sets identical with the set where the function takes its values; to such a function we give the name law of composition.

Alongside these "internal laws", mathematicians have been led (chiefly under the influence of Geometry) to consider another type of "law of composition"; these are the "laws of action", where, besides the set E (which remains as it were in the foreground), an auxiliary set $\Omega$ occurs, whose elements are called operators: the law this time associating with an ordered pair $(\alpha, a)$ consisting of an operator $\alpha \in \Omega$ and an element $a \in E$ a second element $b$ of E. For example a homothety of given centre on the Euclidean space E associates with a real number $k$ (the "homothety ratio", which is here the operator) and a point A of E another point A' of E: this is a law of action on E.

In conformity with the general definitions (Set Theory, IV, § 1, no. 4), being given on a set E one or several laws of composition or laws of action defines a structure on E; for the structures defined in this way we reserve precisely the name algebraic structures and it is the study of these which constitutes Algebra.

There are several species (Set Theory, IV, § 1, no. 4) of algebraic structures, characterized on the one hand by the laws of composition or laws of action which define them and on the other by the axioms to which these laws are subjected. Of course, these axioms have not been chosen arbitrarily, but are just the properties of most of the laws which occur in applications, such as associativity, commutativity, etc. Chapter I is essentially devoted to the exposition of these axioms and the general consequences which follow from them; also there is a more detailed study of the two most important species of algebraic structure, that of a group (in which only one law of composition occurs) and that of a ring (with two laws of composition), of which a field structure is a special case.

In Chapter I are also found the definitions of groups with operators and rings with operators, where, besides the laws of composition, there occur one or several laws of action. The most important groups with operators are modules, of which vector spaces are a particular example, which play an important role both in Classical Geometry and in Modern Analysis. The study of module structures derives its origin from that of linear equations, whence its name Linear Algebra; the results on this subject are to be found in Chapter II.

Similarly, the rings with operators which occur most often are called algebras (or hypercomplex systems). In Chapters III and IV a detailed study is made of two particular types of algebras: exterior algebras, which with the theory of determinants which is derived from them is a valuable tool of Linear Algebra; and polynomial algebras, which are fundamental to the theory of algebraic equations.

In Chapter V there is an exposition of the general theory of commutative fields and their classification. The origin of this theory is the study of algebraic equations in one unknown; the questions which gave birth to this today have little more than a historical interest, but the theory of commutative fields remains fundamental to Algebra, being basic to the theory of algebraic numbers on the one hand and Algebraic Geometry on the other.

As the set of natural numbers has two laws of composition, addition and multiplication, Classical Arithmetic (or Number Theory), which is the study of natural numbers, is subordinate to Algebra. However, related to the algebraic structure defined by these two laws is the structure defined by the order relation "a divides b"; and the distinguishing aspect of Classical Arithmetic is precisely the study of the relations between these two associated structures. This is not the only example where an order structure is thus associated with an algebraic structure by a "divisibility" relation: this relation plays just as important a role in polynomial rings. A general study will therefore be made of this in Chapter VI; this study will be applied in Chapter VII to determining the structure of modules over certain particularly simple rings and in particular to the theory of "elementary divisors".

Chapters VIII and IX are devoted to more particular theories, but which have many applications in Analysis: on the one hand, the theory of semisimple modules and rings, closely related to that of linear representations of groups; on the other, the theory of quadratic forms and Hermitian forms, with the study of the groups associated with them. Finally, the elementary geometries (affine, projective, euclidean, etc.) are studied in Chapters II, VI and IX from a purely algebraic point of view: here there is little more than a new language for expressing results of Algebra already obtained elsewhere, but it is a language which is particularly well adapted to the later developments of Algebraic Geometry and Differential Geometry, to which this chapter serves as an introduction.