3. The evolution of algebra.
There are few notions, in Mathematics, that are more primitive than that of a law of composition: it seems inseparable from the first rudiments of calculation with whole numbers and measurable quantities. The most ancient documents that remain from the Mathematics of the Egyptians and Babylonians show us that they are already in possession of a complete system of rules of calculation for whole numbers , rational numbers , lengths and areas; even though the Babylonian texts that have reached us only deal with problems in which the data have explicit numerical values, they leave no doubt as to the generality attributed to the rules that are used, and denote a quite remarkable technical facility in the handling of equations of the first and second degrees ([232], pp. 179 and following). No hint of a justification of the rules used is found in any case, nor even a precise definition of the operations that occur: these like those remain in the domain of empiricism.
A similar concern, on the other hand, shows itself already very clearly among the Greeks of the classical era; one does not meet yet, it is true, an axiomatic treatment of the theory of whole numbers (such an axiomatisation will not appear until the end of the XIXth century; see p. 24); but, already in the Elements of Euclid, many are the passages giving formal proofs of rules of calculation quite as intuitively "obvious" as those of calculation with whole numbers (for example, the commutativity of the product of two rational numbers). The most remarkable proofs of this kind are those that are connected with the theory of quantities, the most original creation of Greek Mathematics (equivalent, as we know, to our theory of real numbers ; see pp. 148 ff.): Euclid considers there, amongst others, the product of two ratios of quantities, shows that it is independent of the form in which the two ratios are given (first example of a "quotient" of a law of composition
It must not be forgotten that it is only with Viète [319] that the habit is introduced of denoting by letters all the elements (given or unknown) that occur in an Algebra problem. Until then, the only equations which are solved in the treatises on Algebra have numerical coefficients; when the author states a general rule for dealing with analogous equations, he does it (to the best of his ability) in ordinary language; in the absence of an explicit statement of this kind, the carrying out of calculations in the numerical cases treated makes the possession of such a rule more or less likely.
by an equivalence relation), and that it is commutative ([107], Book V, prop. 22-23).
We must not hide, though, the fact that this progress towards rigour is accompanied, in Euclid, with a stagnation, and even some points of regression, where the techniques of algebraic calculations are concerned. The crushing preponderance of Geometry (in the context of which the theory of quantities is manifestly conceived) paralyses all autonomous development of an algebraic notation: the elements occurring in the calculations must, at every moment, be “represented” geometrically: over and above, the two laws of composition which occur are not defined on the same set (addition of ratios is not defined in a general way, and the product of two lengths is not a length but an area); there is, as a result, a lack of flexibility which makes it almost impractical to handle algebraic relations of degree greater than the second.
It is only with the decline of classical Greek Mathematics that one sees Diophantus return to the traditions of the “logistitians” or professional calculators, who had continued to apply as they stood the rules inherited from the Egyptians and Babylonians: not burdening themselves any more with geometrical representations for those “numbers” that he considers, he is naturally led to develop rules for abstract algebraic calculation; for example, he gives rules which (in modern language) are equivalent to the formula for small values (positive or negative) of and ([91 a], v. I, pp. 8-13); a bit further on (pp.12-13) is found the statement of the “rule of signs”, first hint of calculations with negative numbers; finally Diophantus uses, for the first time, a literal symbol to represent an unknown in an equation. On the other hand, he seems barely bothered to connect to general ideas the methods which he applies to the solving of his problems; as to the axiomatic conception of laws of composition, such as was beginning to dawn with Euclid, it appears foreign to the thought of Diophantus as well as to his immediate followers; it will not be found again in Algebra before the beginning of the XIXth century.
It was necessary first, during the intermediary centuries, that on the one hand a system of algebraic notation was developed, adequate for the expression of abstract laws, and that, on the other hand, the notion of “number” received a broadening that allowed, by the observation of sufficiently diversified special cases, an ascent to general concepts. On that point, the axiomatic theory of ratios of quantities, created by the Greeks, was insufficient, because it only made precise the intuitive notion of a real number and the operations on these numbers, already known to the Babylonians in a more confused
Euclid does not give at this point, it is true, a formal definition of the product of two ratios, and the one which is found a bit later in the Elements (Book VI, def. 5) is considered to be interpolated: he has none the less, of course, a perfectly clear conception of this operation and its properties. Diophantus does not know negative numbers; this rule can therefore only be interpreted as relating to calculations with polynomials, and allowing the “expansion” of products such as .
form; it will be a case on the contrary, now, of "numbers" of which the Greeks had not conceived, and of which, at first, no sensible "representation" presented itself: on the one hand, zero and negative numbers, which appeared from the high Middle Ages in Hindu Mathematics; on the other, imaginary numbers, creation of the Italian algebraists in the XVIth century.
If one puts aside the zero, introduced first of all as a symbol of notation before being considered as a number (see p. 45), the common character of these extensions is to be (at least at first) purely "formal". By that it must be understood that the new "numbers" appear first of all as the results of operations applied in situations where they do not have, according to their strict definition, any meaning (for example, the difference of two whole numbers when ): whence the name of "false", "fictitious", "absurd", "impossible", "imaginary", etc. numbers, that have been attributed to them. For the Greeks of the classical era, taken up above all with clear thinking, such extensions were inconceivable; they could only come from calculators more disposed than were the Greeks to have faith even though somewhat mystical in the power of their methods ("the generality of Analysis", as would say the XVIIIth century), and to allow themselves to be dragged along by the mechanics of their calculations without checking at each step its soundness; a faith justified besides most often , by the exact results to which the extensions, to these new mathematical objects, of rules of calculation valid only, in all rigour, for numbers known previously, led. That explains how there is a gradually increasing boldness in considering for themselves (independently of all applications to concrete calculations), these generalisations of the notion of number, which, at first, only arose as intermediaries in a sequence of operations of which the point of departure and the goal were true "numbers"; once this step was taken, more or less tangible interpretations of these new entities began to be sought, which acquired thus the right to be quoted in Mathematics.
On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by the intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representations" for them which are geometric or dynamic. There in any case, together with a progressive improvement in algebraic notation, is the only notable progress in Algebra during the end of the Middle Ages.
Besides this research only constituted a transitory stage in the evolution of the notions under consideration; from the middle of the XIXth century, there has been a return, in a fully conscious way this time, to a formal conception of various extensions of the notion of number, a conception which ended up being integrated in the "formalist" and axiomatic point of view, which dominates the whole of modern mathematics.
At the beginning of the XVIth century, Algebra took new flight, thanks to the discovery, by the mathematicians of the Italian school, of the solution "by radicals" of the equation of the 3rd, then of that of the 4th degree (see pp. 72 ff.); it is on this occasion that, in spite of their repugnance, they found themselves so to speak constrained to introduce into their calculations imaginary numbers; little by little, however, confidence in calculation with these "impossible" numbers is born, as with negative numbers, and that although no "representation" of them was thought up for more than two centuries.
On the other hand, algebraic notation received its decisive improvements from Viète and from Descartes; starting with this latter, algebraic notation is already, with a few exceptions, that which we use today.
From the middle of the XVIIth century to the end of the XVIIIth century, it appears that the vast horizons opened up by the creation of the infinitesimal Calculus made to some extent for a neglect of Algebra in general, and in particular mathematical thought on the laws of composition or on the nature of real and complex numbers. It is thus that the composition of forces and the composition of velocities, well known in Mechanics already from the end of the XVIIth century, did not exert any repercussion on Algebra, although they contained already in embryo vector calculus. Indeed we must await the movement of ideas which, around 1800, leads to the geometrical representation of complex numbers (see p. 161), to see, in pure Mathematics, the use of vector addition.
It is at about the same time that, for the first time in Algebra, the notion of a law of composition extends. in two different directions, to elements that no longer come associated with "numbers" (in the widest meaning given up to that point to this word) except by distant analogies. The first of these extensions is due to C. F. Gauss, at the time of his arithmetical researches on quadratic forms with integral coefficients. Lagrange had defined, in the set of forms with the same discriminant, an equivalence relation, and had, on the other hand, proved an identity which supplied, in this set, a commutative law of composition (not defined everywhere); starting from these results, Gauss shows that this law is compatible with a slightly different equivalence relation ([124 a], v. I, p. 272): "One sees by that", he then says,
We must put on one side the efforts of Leibniz, on the one hand to put in an algebraic form the reasoning of formal logic (see p. 7), on the other hand to set up a "geometric calculus" operating on geometric elements, without the intermediary of co-ordinates ([198 a], v. V, p. 141). But these efforts remained in the nature of outlines, and did not resonate with his contemporaries; they were not to be taken up again until during the XIXth century (see below).
This operation is in any case introduced without any reference to Mechanics; the link between the two theories is only recognised explicitly by the founders of vector Calculus, in the second third of the XIXth century (see p. 61).
Two forms are equivalent when one of them can be deduced from the other by a "change of variables" , where are integers such that .
"what should be understood by a class composed from two or more classes". He proceeds afterwards to the study of the "quotient" law that he has thus just defined, establishes in substance that it is (in modern language) an abelian group law, and this by arguments of which the generality surpasses by far, in the majority of cases, the special case that Gauss considers (for example, the argument by which he proves the uniqueness of the symmetric element can be applied to any law of composition whatever (ibid., p. 273)). But he does not stop there: returning a little later to the question, he recognises the analogy between the composition of classes and the multiplication of integers modulo a prime number (ibid. p. 371), but notes also that the group of classes of quadratic forms of given discriminant is not always a cyclic group; the indications that he gives on this matter make us think that he might perhaps have realised, at least in this particular case, the general structure of finite abelian groups ([124 a], v. I, p. 374 and v. II, p. 266).
The other series of researches of which we wish to speak ends, it also, with the notion of group, thus introducing it in a definitive way into Mathematics: it is the "theory of substitutions", a development of the ideas of Lagrange, Vandermonde and Gauss on the solution of algebraic equations. We will not give here a detailed history of this question (see pp. 75 ff.); we must remember the definition by Ruffini [265], then Cauchy ([56 a], (2), v. I, p. 64), of the "product" of two permutations of a finite set, and the first notions about finite groups of permutations: transitivity, primitivity, neutral element, commuting elements, etc.. But these first researches remain, on the whole, fairly superficial, and it is Evariste Galois who must be considered the real initiator of the theory: having, in his memorable work [123], reduced the study of algebraic equations to that of the groups of permutations that he associates with them, he deepens considerably this latter, as much in its concern with general properties of groups (it is Galois who first defines the notion of normal subgroup and recognises its importance), as in the determination of groups possessing particular properties (where the results that he obtains are classed still today amongst the most subtle of the theory). It is also to
It is quite remarkable that Gauss denotes the composition of classes of quadratic forms additively, in spite of the analogy that he points out himself, and in spite of the fact that the identity of Lagrange, who defines the composition of two forms, suggests much more naturally multiplicative notation (to which in any case all the successors of Gauss returned). One must see in this indifference to matters of notation a further witness to the generality to which Gauss had certainly arrived in his conceptions relating to laws of composition. He did not limit in any case his views to commutative laws, as is shown by a fragment not published during his lifetime, but dating from the years 1819-1820, where he gives, more than twenty years before Hamilton, the formulae for the multiplication of quaternions ([124 a], v. VIII, p. 357).
The notion of composite function was naturally known well before, at least for functions of real or complex variables; but the algebraic aspect of this law of composition, and the link with the product of two permutations, do not come to light until the work of Abel ([1], v. I, p. 478) and of Galois.
Galois that the first idea of the "linear representation of groups" is due, and this fact proves clearly that he was in possession of the notion of the isomorphism of two group structures, independently of their "realisations".
However, if it appears incontestable that the brilliant methods of Gauss and Galois had led them to a very wide conception of the law of composition, they did not have the opportunity to develop particularly their ideas on this point, and their work did not have any immediate action on the evolution of abstract Algebra. It is in a third direction that the clearest progress towards abstraction is made: following reflections on the nature of imaginaries (of which the geometrical representation had caused, at the beginning of the XIXth century, fairly substantial work), the algebraists of the English school bring out first, from 1830 to 1850, the abstract notion of a law of composition, and enlarge immediately the field of Algebra by applying this notion to a host of new mathematical objects: the algebra of Logic with Boole (see p. 8), vectors, quaternions and general hypercomplex systems with Hamilton [145 a], matrices and non-associative laws with Cayley ([58], v. I, pp. 127 and 301, and v. II, pp. 185 and 475). A parallel evolution takes place independently on the continent, notably as far as vector Calculus is concerned (Möbius, Bellavitis), linear Algebra and hypercomplex systems (Grassmann) (see pp. 61).
From all this bubbling up of original and fruitful ideas that come to revive Algebra in the first half of the XIXth century, this latter emerges renewed even in its trends. Before, methods and results gravitated around the central problem of the solution of algebraic equations (or diophantine equations in Number Theory): "Algebra", says Serret in the Introduction to his Higher Algebra Course [284], "*is, properly speaking, the Analysis of equations*". After 1850, even if the treatises on Algebra still leave for a long time the preeminence to the theory of equations, new research is no longer dominated by the concern for immediate applications to the solution of numerical equations, and is more and more oriented towards what we would now consider as the essential problem of Algebra, the study of algebraic structures as such.
This work is divided fairly neatly into three streams, which extend respectively the three movements of ideas that we have analysed above, and which are pursued in parallel without mutual measurable influence until the last years of the XIXth century.
It is in this instance that Galois, by a bold extension of the "formalism" that had led to complex numbers, considers "imaginary roots" of a congruence modulo a prime number, and discovers thus finite fields (see p. 82).
That of Galois remained in any case ignored until 1846, and that of Gauss only exerted a direct influence on Number Theory.
The main theories developed during this time are remarkably displayed in the contemporary work of H. Hankel [146], where the abstract notion of a law of composition is conceived and presented with perfect clarity.
We leave voluntarily on one side here all that concerns, during this period, the evolution of algebraic geometry, and the theory of invariants, which is tightly linked
It is first the building up, by the German school of the XIXth century (Dirichlet, Kummer, Kronecker, Dedekind, Hilbert) of the theory of algebraic numbers, coming out of the work of Gauss, to whom is due the first study of this kind, that of numbers ( and rational). We will not follow here the evolution of this theory: we must simply pick up, for our purposes, the abstract algebraic notions that emerge there. Already from the first successors of Gauss, the idea of a field (of algebraic numbers) is at the base of all the work in question (as are also the researches of Abel and Galois on algebraic equations); its field of application grows when Dedekind and Weber [80] copy the theory of algebraic functions of one variable onto that of algebraic numbers. Also due to Dedekind ([79], v. III, p. 251) is the introduction of the notion of ideal, which provides a new example of a law of composition between sets of elements; it is to him and Kronecker that the ever greater role played by abelian groups and modules in the theory of algebraic fields goes back; we will return to this later on (see pp. 63 ff. and 93 ff.).
We leave also till later the history of the development of linear Algebra (p. 57) and hypercomplex systems (p. 117), which take place without introducing any new algebraic notion during the end of the XIXth century and the beginning of the XXth century, in England (Sylvester, W. Clifford) and in America (B. and C. S. Peirce, Dickson, Wedderburn) following the route laid out by Hamilton and Cayley, in Germany (Weierstrass, Dedekind, Frobenius, Molien) and in France (Laguerre, E. Cartan) in a manner independent of the Anglo-Saxons and following fairly different methods.
As for the theory of groups, it is mainly under the form of the theory of finite permutation groups that it is developed at first, following the publication of the works of Galois and their diffusion by the works of Serret [284], and above all the great "Treatise on Substitutions" of C. Jordan [174 a]. This latter summarises there, perfecting them greatly, the work of his predecessors on the particular properties of permutation groups (transitivity, primitivity, etc.), obtaining results of which the greater part have hardly been surpassed since; he studies as well, in a deep way, very important particular groups, the linear groups and their subgroups (see p. 138); as well, it is he who introduces the fundamental notion of the representation of one group by another, as well as (a little later) that of quotient group, and who proves part of the theorem known by the name of "Jordan-Hölder Theorem". It is finally to Jordan that the first study of infinite groups goes back [174 b], that S. Lie on the one hand, F. Klein and H. Poincaré on the other, were to develop considerably, in two different directions, a few years later.
Jordan had only established the invariance (up to order) of the orders of the quotient groups of a "Jordan-Hölder series" for a finite group; it is O. Hölder who showed that the quotient groups themselves were (up to order) independent of the series under consideration [165].
In the meantime, it had gradually become obvious that what is essential in a group is its law of composition and not the nature of the objects making up the group (see for example [58], v. II, pp. 123 and 131 and [79], v. III, p. 439). However, even the researches on finite abstract groups are still for a long time conceived as studies about groups of permutations, and it is only around 1880 that there is a conscious development of the autonomous theory of finite groups. We cannot follow further the history of this theory; we limit ourselves to mentioning two of the tools which still today are amongst the most used in the study of finite groups, and which both go back to the XIXth century: the theorems of Sylow on -groups, which date from 1872 [303] and the theory of characters, created in the last years of the century by Frobenius ([119], v. III, pp.1-37). We send the reader wishing to deepen his knowledge of the theory of finite groups and the numerous and difficult problems it gives rise to, to the monographs [44 b], [131], [291] and [341].
It is towards 1880 also that presentations of groups begin to be studied systematically; before that, only presentations of particular groups had been studied, for example of the alternating group in the work of Hamilton [145 b], or of the monodromy groups of differential equations and Riemann surfaces (Schwarz, Klein, Schlafli). W. Dyck was the first [97] to define (without giving it a name) the free group generated by a finite number of generators, but he is less interested in it for itself than as a "universal" object allowing the precise definition of what is meant by a group given "by generators and relations". Developing an idea first put out by Cayley, and visibly influenced elsewhere by the work of the school of Riemann mentioned above, Dyck describes an interpretation of a group with a given presentation, where each generator is represented by a product of two inversions with respect to tangent circles or secants (see also [44 b], chap. XVIII). A little later, after the development of the theory of automorphic functions by Poincaré and his successors, then the introduction by Poincaré of the tools of algebraic Topology, the first studies of fundamental groups will go hand in hand with that of group presentations (Dehn, Tietze), the two theories providing mutual support. It is in any case a topologist, J. Nielsen who, in 1924 introduced the terminology free group in the first deep study of its properties [234]; almost immediately afterwards E. Artin (still in connection with topological questions) introduces the notion of free product of groups, and O. Schreier defines more generally the free product with amalgamated subgroup. Here again, we cannot tackle the history of subsequent developments in this direction, referring to the work [216] for more details.
It is not either the place to speak of the extraordinary prosperity that has been known, since the end of the XIXth century, by the idea of group (and that of invariant, to which it is intimately linked) in Analysis, in Geometry, in Mechanics and in theoretical Physics. It is by an analogous invasion of
The existence of a subgroup of order in a finite group whose order is divisible by is mentioned without proof in the papers of Galois ([123], p. 72).
this notion, and of the algebraic notions that are related to it (groups with operators, rings, ideals, modules) in those parts of Algebra which until then appeared fairly remote from their proper domain, that the last period of the evolution that we are retracing here is marked, and which ends with the synthesis of the three tendencies that we have followed above. This unification is above all the work of the modern German school: begun with Dedekind and Hilbert in the last years of the XIXth century, the work of axiomatisation of Algebra was vigorously pursued by E. Steinitz [294 a], then, from 1920, under the impulsion of E. Artin, E. Noether and the algebraists of their schools (Hasse, Krull, O. Schreier, van der Waerden). The treatise of van der Waerden [317 a], published in 1930, gathered together for the first time these works in a complete exposition, opening the way and serving as guide to the multiple subsequent researches into abstract Algebra.