4. Linear algebra and multilinear algebra.

Linear algebra is at the same time one of the oldest branches of mathematics and one of the newest. On the one hand, problems are found in the origins of mathematics which are solved by a single multiplication or division, that is to say by the calculation of a value of a function $f(x) = ax$, or by the solution of an equation $ax = b$: those are typical problems of linear algebra, and it is not possible to deal with them, nor even to state them correctly, without "thinking linearly".

On the other hand, not only these questions, but almost everything that touches on equations of the first degree, had for a long time been relegated to elementary teaching, when the modern development of the notions of field, ring, topological vector space, etc. ..., came to free and give importance to the essential notions of linear algebra (for example duality); it is then that the essentially linear character of almost all modern algebra was realised, of which this "linearisation" is even one of the distinguishing traits, and that the place that is its due was given back to linear algebra. To give its history, from the point of view where we put ourselves, will thus be a task that is as important as it is difficult; and we will have to be content here with fairly summary indications.

From what comes earlier, it follows that linear algebra has without doubt been born in order to supply the needs of practising calculators; it is thus that we see the rule of three ([311], v. I, pp. 150-155) and the rule of false position, more or less clearly stated, playing an important role in all the manuals of practical arithmetic, from the Rhind papyrus of the Egyptians up to those which are in use in our primary schools, passing through Aryabhata, the Arabs, Leonardo of Pisa and the innumerable "books of calculation" of the Middle Ages and of the Renaissance; but they have perhaps never been anything other than an extract, for the use of practitioners, of more advanced scientific theories.

As for actual mathematicians, the nature of their research on linear algebra is a function of the general structure of their science. Ancient Greek mathematics, as it is set out in the Elements of Euclid, has developed two abstract theories of a linear character, on the one hand that of quantities ([107], book V, cf. p. 152), on the other hand that of whole numbers ([107], book VII). With the Babylonians, we find methods that are much closer to our elementary algebra; they know how to solve, and in a very elegant way ([232], pp. 181-183) systems of equations of the first degree. For a very long time, however, progress in linear algebra was mainly tied to that of algebraic calculation, and it is under this aspect, foreign to the present Note, that it is reasonable to consider it; to reduce, indeed, a linear system to an equation of the type $ax = b$, it is enough, if only one unknown is involved, to know the rules (already substantially stated by Diophantus) by which one can move terms from one member to another, and combine similar terms; and, if many unknowns are involved, it is enough to know as well how to eliminate them successively until only one remains. So the treatises on algebra, until the XVIIIth century, were thought to have done everything as far as the first degree is concerned, once they had laid out these rules; as for a system with as many unknowns as equations (they do not consider any others) where the first members would not be linearly independent forms, they content themselves invariably with noting in passing that that indicates a badly posed problem. In the treatises of the XIXth century, and even certain more recent works, this point of view has only been changed by the progress of notation, which allows the writing of systems of $n$ equations in $n$ unknowns, and by the introduction of determinants, which allow the statement of formulae for the explicit solution in the "general case"; this progress, for which the credit would be due to Leibniz ([198 a], v. I, p. 239) if he had published his ideas on the subject, is due principally to mathematicians of the XVIIIth and the beginning of the XIXth century.

But we must first consider several streams of ideas which, much more than the study of systems of linear equations, have contributed to the development of linear algebra in the sense in which we understand it. Inspired by the study of Apollonius, Fermat ([109], v. I, pp. 91-110; French translation, v. III, pp. 85-101), conceives, even before Descartes ([85 a], v. VI), the principles of analytic geometry, has the idea of the classification of plane curves by means of their degree (which, having become more and more familiar to all mathematicians, can be considered as definitely acquired towards the end of the XVIIth century), and states the fundamental principle of an equation of the first degree, in the plane, representing a straight line, and an equation of the second degree a conic: a principle from which he deduces straightaway some "very pretty" consequences concerning geometric loci. At the same time, he announces ([109], v. I, pp. 184-188; French translation, v. III, pp. 159-163) the classification of problems as problems solved, problems which reduce to an equation in two unknowns, to an equation in three unknowns, etc.; and he adds: the first consist in determining a point, the second a line or plane locus, those following a surface; etc. ("...such a problem does not seek only a point or a line, but a whole surface proper to the question; it is from that that are born the superficial loci, and so on for those following", loc. cit. p. 186; there can be found already the embryo of geometry in $n$ dimensions). This piece of writing, stating the principle of dimension in algebra and in geometric algebra, indicates a fusion of algebra and geometry, absolutely in line with modern ideas, but which, as has been seen already, took more than two centuries to penetrate minds.

At least these ideas ended up soon in the flowering of analytic geometry, which acquires its whole range in the XVIIIth century with Clairault, Euler, Cramer, Lagrange and many others. The linear character of the formulae for the transformation of co-ordinates in the plane and in space, which cannot fail to have been noticed already by Fermat, is put into prominence for example by Euler ([108 a], (1), v. IX, chap. II-III, and Append. chap. IV), who builds on this the classification of planar lines, and that of surfaces, according to their degree (invariant precisely because of the linearity of these formulae); it is also he (loc. cit., chap. XVIII) who introduces the word "affinity" to indicate the relation between curves that can be deduced from one another by a transformation $x' = ax, y' = by$ (but without noticing anything of a geometric invariant in this definition which remains attached to a particular choice of axes). A bit later we see Lagrange ([191], v. III, pp. 661-692) devoting a whole memoir, which will remain justly famous for a long time, to typically linear and multilinear problems of analytical geometry in three dimensions. It is at about the same time, in connection with the linear problem consisting of the search for a plane curve passing through given points, that the notion of determinant takes shape, first in a somewhat empirical manner, with Cramer [72] and Bézout [21]; developed afterwards by several authors, this notion and its essential properties are definitively brought out by Cauchy ([56 a], (2), v. I, pp. 91-169) and Jacobi ([171], v. III, pp. 355-392).

On the other hand, while mathematicians had some slight tendency to hold first degree equations in some disdain, the solution of differential equations constituted on the contrary a major problem; it was natural that, amongst these equations, one picked out at an early stage the linear equations, with constant coefficients or not, and that their study contributed to enhance the value of linearity and that which goes with it. That is just what is seen with d'Alembert [75 c], Lagrange ([191], v. I, pp. 471-668) and Euler ([108 a], (1), v. XII); but only the first of these judges it useful to say that the general solution of a non homogeneous equation is the sum of a particular solution and of the general solution of the corresponding homogeneous equation; moreover, when these authors state that the general solution of the linear homogeneous equation of order $n$ is a linear combination of $n$ particular solutions, they do not add that these must be linearly independent, and make no effort to render explicit this latter notion; on this point as on so many others, it is only, it appears, the teaching of Cauchy at the École Polytechnique that will throw some light ([56 b], pp. 573-574). But already Lagrange (loc. cit.) introduces also (solely for calculations, it is true, and without giving it a name) the adjoint equation $L^*(y) = 0$ of a linear differential equation $L(y) = 0$, a typical example of duality by virtue of the relation

valid when $y$ and $z$ are zero at the ends of the interval of integration; more precisely, and 30 years before Gauss had explicitly defined the transpose of a linear substitution in 3 variables, we see there without doubt the first example of a "functional operator" $L^*$ the transpose or "adjoint" of an operator $L$ given by means of a bilinear function (here the integral $\int yz dx$).

At the same time, and with Lagrange also ([191], v. III, pp. 695-795), linear substitutions, in 2 or 3 variables at first, were on the way to conquer arithmetic. It is clear that the set of values of a function $F(x, y)$, when $x$ and $y$ take all integer values, does not change when an arbitrary linear substitution, with integer coefficients, of determinant 1, is performed on $x, y$; it is on this fundamental observation that Lagrange bases the theory of the representation of numbers by forms, and that of the reduction of forms; and Gauss, by means of a step of which it has become difficult for us to appreciate all the boldness, gets out of it the notion of equivalence and that of a class of forms (cf. p. 50); on this point, he recognises the necessity of some elementary principles relative to linear substitutions, and introduces in particular the notion of transpose or adjoint ([124 a], v. I, p. 304). From that moment on, the arithmetic study and the algebraic study of quadratic forms in 2, 3 and later $n$ variables, that of bilinear forms that are closely linked to them, and more recently the generalisation of these notions to an infinity of variables, was, up to our time, to form one of the most fruitful sources of progress for linear algebra (cf. pp. 136 ff.).

But, a still more decisive progress perhaps, Gauss, in these same Disquisitiones, created (cf. p. 51) the theory of finite abelian groups, which arises in four different ways, by means of the additive group of whole numbers modulo $m$ (for $m$ an integer ), by means of the multiplicative group of numbers prime to $m$ modulo $m$, by means of the group of classes of binary quadratic forms, and finally by means of the multiplicative group of $m$th roots of unity; and, as we have already noted, it is clearly as abelian groups, or to put it better as modules over $\mathbf{Z}$, that Gauss treats all these groups, studies their structure, their isomorphism relations, etc.. In the module of "complex integers" $a + bi$, he is seen later studying an infinite module over $\mathbf{Z}$, of which he has no doubt not missed seeing the isomorphism with the module of periods (discovered by him in the complex domain) of elliptic functions; in any case this idea appears already clearly with Jacobi, for example in the famous proof of the impossibility of a function with 3 periods and in his views on the problem of inversion of abelian integrals ([171], v. II, pp. 25-50), to end up soon with the theorems of Kronecker ([186 a], v. III_1, pp. 49-109).

Here, in streams of which we have sought to follow the thread and sometimes the meanderings, another comes to join in, which for a long time had remained underground. As it will be brought out elsewhere in more detail (see p. 130), "pure" geometry in the sense in which it had been understood in the previous century, that is to say essentially the projective geometry of the plane and of space without the use of co-ordinates, had been created in the XVIIth century by Desargues [84], whose ideas, appreciated at their true worth by a Fermat, put into use by a Pascal, had subsequently fallen into disuse, eclipsed by the brilliant progress of analytical geometry; it is brought back to pride of place towards the end of the XVIIIth century, by Monge, then Poncelet and his emulators, Brianchon, Chasles, sometimes completely and voluntarily separated from analytical methods, sometimes (especially in Germany) mixed intimately with them. Yet projective transformations, from whatever point of view they are considered (synthetic or analytical), are none other of course than linear substitutions on the projective or "barycentric" co-ordinates; conics (in the XVIIth century), and later quadrics, of which the projective theory has for a long time been the principal object of study of this school, are none other than quadratic forms, of which we have already brought out above the close connection with linear algebra. To these notions is joined that of polarity: created, it also, by Desargues, the theory of poles and polars becomes in the hands of Monge and his successors, and soon under the name of principle of duality, a powerful instrument of transformation for geometric theorems; if one does not dare affirm that its links with adjoint differential equations were noticed only belatedly (they are indicated by Pincherle at the end of the century), what has none the less not been missed, Chasles bears testimony to it ([60 b], p. 55), is to notice its kinship with the notion of reciprocal spherical triangle, introduced into spherical trigonometry by Viète ([319], p. 418) and Snellius already in the XVIth century. But duality in projective geometry is only one aspect of the duality of vector spaces, taking account of the modifications that are imposed by passing from an affine space to a projective space (which is a quotient space of it, by the relation "scalar multiplication").

The XIXth century, more than any period in our history, was rich in mathematicians of the first rank; and it is difficult in a few pages, even limiting oneself to the most outstanding aspects, to describe all that the fusion of these movements of ideas produces in their hands. Between the purely synthetic methods on the one hand, a kind of Procrustean bed where their orthodox adepts torture themselves, and the analytic methods linked to a system of co-ordinates arbitrarily inflicted on space, one feels soon the need for a kind of geometrical calculus, dreamed of but not created by Leibniz, imperfectly outlined by Carnot [51]: it is first the addition of vectors that appears, implicit in Gauss in his geometrical representation of imaginary numbers and the application that he makes of it to elementary geometry (cf. p. 162), developed by Bellavitis under the name of "method of equipollence", and that takes its definitive shape in Grassmann, Möbius, Hamilton; at the same time, under the name of "barycentric calculus", Möbius gives a version of it adapted to the needs of projective geometry ([223], v. I).

At the same time, and amongst the same men, the passage takes place, so natural (as soon as this path has been entered) that we have seen it already announced by Fermat, from the plane and "ordinary" space to space of $n$ dimensions; a passage that is even inevitable, since algebraic phenomena, which are interpreted for 2 or 3 variables as themselves in geometric terms, subsist without change for any number of variables; to impose therefore, in the use of geometric language, a limitation to 2 or 3 dimensions, would be for the modern mathematician a yoke as inconvenient as that which continually prevented the Greeks from extending the notion of number to ratios of incommensurable quantities. Also the language and ideas relating to space of $n$ dimensions appear almost simultaneously on all sides, obscurely in Gauss, clearly among the mathematicians of the subsequent generation; and their more or less great measure of boldness in using it took less account of their mathematical leanings than of their philosophical views or even just practical considerations. In any case, Cayley and Grassmann, in about 1846, manipulate these concepts with the greatest of ease (and that, says Cayley differing from Grassmann ([58], v. I, p. 321), "without recourse to any metaphysical notion"); in Cayley, one remains constantly very near to the analytic interpretation and co-ordinates, whereas in Grassmann it is from the first, with the addition of vectors in space of $n$ dimensions, the geometric aspect that takes the lead, to end up with developments of which we shall speak in a moment.

However, the impulsion that came from Gauss was pushing mathematicians, in two different ways, towards the study of algebras or hypercomplex systems. On the one hand, one could not avoid trying to extend the domain of real numbers other than by the introduction of the "imaginary unit" $i = \sqrt{-1}$, and to open up thus perhaps domains vaster and more fruitful than that of the complex numbers. Gauss convinced himself ([124 a], v. II, p. 178) of the impossibility of such an extension, at least if one seeks to keep the principal properties of the complex numbers, that is to say in modern language, those that make it a commutative field; and, either because of his influence, or independently, his contemporaries seem to have shared this conviction, which will only be justified much later by Weierstrass ([329 a], v. II, pp. 311-332) in a precise theorem. But, as soon as one interprets the multiplication of complex numbers as rotations in the plane, one is led, if intending to extend this idea to space, to envisage (since rotations in space form a non abelian group) non commutative multiplications; it is that which is one of the ideas that guides Hamilton$^1$ in his discovery of quaternions [145 a], the first example of a non commutative field. The singularity of this example (the only one, as Frobenius was to prove later, that can be constructed on the field of reals) restricted somewhat its scope, in spite or perhaps even because of the formation of a school of fanatical "quaternionists": a strange

$^1$ Cf. the interesting preface to his Lectures on quaternions [145 a] where he retraces the history of his discovery.

phenomenon, which repeats itself later around the work of Grassmann, then the popularisers who draw out of Hamilton and Grassmann what has been called the “vector calculus”. The abandonment a little later of associativity, by Graves and Cayley who construct the “Cayley numbers”, does not open up an interesting vein. But, after Sylvester had introduced matrices, and (without giving it a name) had clearly defined rank ([304], v. I, pp. 145-151), it is again Cayley ([58], v. II, pp. 475-496) who creates its calculus, not without noting (an essential fact, often lost sight of later on) that a matrix is only an abbreviated notation for a linear substitution, just as in fact Gauss denoted by $(a, b, c)$ the form $aX^2 + 2bXY + cY^2$. What is there is besides only one of the aspects, the most interesting for us no doubt, of the abundant production by Sylvester and Cayley on determinants and all that is linked with them, a production bristling with ingenious identities and impressive calculations.

It is also (among other things) an algebra over the reals that Grassmann discovers, the exterior algebra to which his name remains linked. His work, prior even to that of Hamilton ([134], v I_1), created in a moral solitude which was almost complete, remained for a long time badly known, because no doubt of its originality, because also of the philosophical fog by which it begins to be shrouded, and which for example turns Möbius away at first. Furnished with concerns analogous to those of Hamilton but with a wider range (and which, as he realises soon, were those themselves of Leibniz), Grassmann constructs a vast algebraico-geometric edifice, resting on a geometric or “intrinsinc” conception (already just about axiomatised) of the vector space of $n$ dimensions; amongst the most elementary of the results which he ends up with, let us quote for example the definition of linear independence of vectors, that of dimension and the fundamental relation $dim\ V + dim\ W = dim\ (V + W) + dim\ (V \cap W)$ (loc. cit., p. 209; cf. [134], v. I_2, p. 21). But it is above all the exterior multiplication, then the interior multiplication, of multivectors, that provide him with the tools by means of which he deals easily with, first the problems of linear algebra proper, then those which are concerned with the Euclidean structure, that is to say with the orthogonality of vectors (where he finds the equivalent of the duality which he is missing).

The other path opened up by Gauss, in the study of hypercomplex systems, is that which starts with the complex integers $a + bi$; from these, one carries on quite naturally to more general algebras or hypercomplex systems, over the ring $\mathbf{Z}$ of integers or over the field $\mathbf{Q}$ of rationals, and first of all to those foreseen already by Gauss which are generated by the roots of unity, then to the fields of algebraic numbers and to modules of algebraic integers: those are the principal objects of the work of Kummer, the study of the former is tackled by Dirichlet, Hermite, Kronecker, Dedekind. Here, contrary to what happens with algebras over the reals, it is not necessary to give up any of the characteristic properties of commutative fields, and it is to these that, during the whole of the XIXth century, work is limited. But the linear properties, and for example the search for a basis for the integers of the field (indispensable for a general definition of the discriminant) play in many respects an essential role; and, in Dedekind in any case, the methods are destined to become typically "hypercomplex"; besides Dedekind himself, without addressing in a general way the problem of algebras, is aware of this aspect of his work, and of that which ties it for example to the results of Weierstrass on hypercomplex systems over the reals ([79], in particular vol. II, pp. 1-19). At the same time the determination of the structure of the multiplicative group of units in a field of algebraic numbers, carried out by Dirichlet in his famous notes ([92], v. I, pp. 619-644) and almost at the same time by Hermite ([159], v. I, pp. 159-160), was eminently suitable to clarify the ideas about modules over $\mathbf{Z}$, their systems of generators, and, when they have them, their bases. Then the notion of ideal, that Dedekind defines in algebraic number fields (as a module over the ring of integers of the field) ([79], v. III, p. 251), whereas Kronecker introduces in polynomial rings (under the name of "system of modules") an equivalent notion ([186 a], v. II, pp. 334-342), gives the first examples of modules over rings more general than $\mathbf{Z}$; and in the same authors, then in Hilbert, is released little by little in particular cases the notion of a group with operators, and the possibility of always constructing, starting with such a group, a module over a suitably defined ring.

At the same time, the arithmetico-algebraic study of quadratic and bilinear forms, and their "reduction" (or, what amounts to the same, that of matrices and their "invariants") leads to the discovery of general principles concerning the solution of systems of linear equations, principles that, for lack of the notion of rank, had escaped Jacobi.$^2$ The problem of the solution in integers of systems of linear equations with integral coefficients is tackled and solved, at first in a special case by Hermite, then in all its generality by H. J. Smith ([287], v. I, pp. 367-409); the results of this latter are found again, only in 1878, by Frobenius, within the framework of a vast programme of research instituted by Kronecker, and in which Weierstrass also participates; it is incidentally Kronecker, in the course of his work, who gives the definitive form to the theorems on linear systems with real (or complex) coefficients, which is also elucidated in an obscure manual, with the scrupulous care which characterises him, by the famous author of Alice in Wonderland; as for Kronecker, he disdains publishing his results, and leaves it to his colleagues and disciples; even the word rank is only introduced by Frobenius. It is also in their teaching at the university of Berlin that Kronecker [186 b] and Weierstrass introduce the "axiomatic" definition of determinants (as a multilinear alternating function of $n$ vectors in space of $n$ dimensions, normalised so as to take the value 1 for the identity matrix), a definition equivalent to that which

$^2$ On the classification of systems of $n$ equations in $n$ unknowns when the determinant is zero, he says ([171], v. III, p. 370): "paullo prolixum videtur negotium" (it will not be possible to elucidate it briefly).

can be deduced from the calculations of Grassmann; it is in his lectures that Kronecker still, without feeling the need to give it a name, and in a still not intrinsic form, introduces the tensor product of spaces and the "Kroneckerien" product of matrices (a linear substitution induced, on a tensor product, by given linear substitutions applied to the factors).

These researches should not be separated either from invariant theory, created by Cayley, Hermite and Sylvester (the "invariant trinity" of whom later Hermite speaks in his letters) and which, from a modern point of view, is above all a representation theory of linear groups. There emerges, as an algebraic equivalent of the duality of projective geometry, the distinction between series of cogredient and contragredient variables, that is to say vectors in a space and vectors in the dual space; and after attention had been drawn first to forms of low degree, then of arbitrary degree, in 2 or 3 variables, little time is lost in examining bilinear, then multilinear forms, in several series of "cogredient" or "contragredient" variables, which is equivalent to the introduction of tensors; this becomes conscious and is popularised when, under the inspiration of invariant theory, Ricci and Levi-Civitá, in 1900, introduce the "tensor calculus" into differential geometry [257], which later was very much in fashion following its use by "relativist" physicists. It is still the progressive interpenetration of invariant theory, of differential geometry, and of the theory of partial differential equations (especially the problem of Pfaff so called, and its generalisations) which brings little by little geometers to consider, first of all bilinear alternating forms of differentials, in particular the "bilinear covariant" of a form of degree 1 (introduced in 1870 by Lipschitz, then studied by Frobenius), to finish with the creation, by E. Cartan ([52 a], v. II₁, pp. 303-396) and Poincaré ([251 b], v. III, chap. XXII) of the calculus of exterior differential forms. Poincaré introduces these, in the context of the formation of his integral invariants, as being expressions which appear in multiple integrals, whereas Cartan, guided no doubt by his researches on algebras, introduces them in a less formal way, but not without observing also that the algebraic part of their calculus is identical to the exterior multiplication of Grassmann (hence the name he adopts), thereby putting definitively into its true place the latter's work. The translation into the notation of tensor calculus, of exterior differential forms, shows besides immediately their link with asymmetric tensors, which will, as soon as one returns to a purely algebraic point of view, make it to be seen that they are to alternating multilinear forms what covariant tensors are to arbitrary multilinear forms; this aspect becomes clearer still with the modern theory of representations of linear groups; and by that is recognised, for example, the substantial identity between the definition of determinants given by Weierstrass and Kronecker, and that which results from the calculus of Grassmann.

We reach thus the modern era, where the axiomatic method and the notion of structure (felt at first, defined only recently), allow us to separate concepts that up till then had been inextricably mixed, to formulate that which was vague or subconscious, to prove in the generality which is appropriate the theorems which had only been known in particular cases. Peano, one of the creators of the axiomatic method, and one of the first mathematicians also to appreciate at its true worth the work of Grassmann, gives already in 1888 ([246 b], chap. IX) the axiomatic definition of vector spaces (whether of finite dimension or not) over the field of reals, and, with a fully modern notation, linear maps from such a space to another; a little later, Pincherle seeks to develop applications of linear algebra, thus conceived, to the theory of functions, in a direction, it is true, which has hardly shown itself to be fruitful; at least his point of view will allow him to recognise in the "Lagrange adjoint" a particular case of the transposition of linear maps: which appears soon, still more clearly, and for partial differential equations as well as for differential equations, in the course of the memorable work of Hilbert and his school on Hilbert space and its applications to analysis. It is in connection with these latter researches that Toeplitz [309 a], also introducing (but by means of co-ordinates) the most general vector space over the reals, makes the fundamental observation that the theory of determinants is useless for the proof of the main theorems of linear algebra, which allows the extension of these without difficulty to infinite dimensional spaces; and he indicates also that the linear algebra thus understood is applicable of course to any commutative base field.

On the other hand, with the introduction by Banach, in 1922, of the spaces which carry his name,$^3$ one meets, it is true in a problem which is topological as much as algebraic, spaces which are not isomorphic to their dual (see pp. 214 ff.). Already between a vector space and its dual, there is no "canonical" isomorphism, that is to say determined by the structure, which had been reflected for a long time by the distinction between cogredient and contragredient. It seems nonetheless that there is no doubt that the distinction between a space and its dual was only definitively established following the work of Banach and his school; it is this work too which raised an interest in the notion of codimension. As for the duality or "orthogonality" between vector subspaces of a space and those of its dual, the way in which it is formulated today demonstrates an analogy which is not just coincidental with the modern formulation of the main theorem of Galois theory, and with the duality of Pontrjagin so called in locally compact abelian groups; this goes back to Weber, who, in the course of arithmetic researches, stated the bases of it in 1866 for finite groups [327 d]; in Galois theory, the "duality" between subgroups and subfields takes shape with Dedekind and Hilbert; and the orthogonality between vector subspaces is visibly derived, first of all from the duality between linear varieties in projective geometry, and also from the notion and properties of completely orthogonal varieties in a Euclidean space or a Hilbert space (whence its name). All these threads gather together in the

$^3$ They are normed complete vector spaces, over the field of real numbers or that of complex numbers.

contemporary era, in the hands of algebraists such as E. Noether, Artin and Hasse, of topologists such as Pontrjagin and Whitney, not without mutual influence being exerted between one group and the other.

At the same time a critical examination is being undertaken, which is meant to eliminate, at each point, the hypotheses which are not truly indispensable, and especially those which would prevent certain applications. Thus is realised the possibility of substituting rings for fields in the notion of vector space, and, creating the general notion of module, treating at one and the same time these spaces, abelian groups, particular modules already studied by Kronecker, Weierstrass, Dedekind, Steinitz, and even groups with operators, and for example to apply to them the Jordan-Hölder theorem; at the same time the transition to non commutativity is made, by means of the distinction between right modules and left modules, to which the modern development of the theory of algebras by the American school (Wedderburn, Dickson) and especially the German school (E. Noether, Artin) was leading.

Finally there has emerged, quite recently, the last of the tendencies of which we must speak here: contained in embryo in the theorem of Dedekind ([79], vol. III, p. 29) according to which arbitrary automorphisms of a commutative field are always linearly independent, the linearisation of Galois theory is carried out with Artin [7 a], then it is soon generalised to arbitrary extensions of commutative fields, and even to extensions of non commutative fields ([172 d], chap. VII).