Historical note

(Chapters VI and VII)

(N.B. — Roman numerals in parentheses refer to the bibliography at the end of this note.)

Elementary arithmetical operations, in particular operations with fractions, inevitably lead to a number of empirical observations about divisibility between integers. But neither the Babylonians (although they were such experts in algebra), nor the Egyptians (despite their acrobatic skill at manipulating fractions) seem to have been acquainted with the general rules governing these properties, and the initiative in this respect is due to the Greeks. Their arithmetical work, a masterly exposition of which can be found in Books VII and IX of Euclid (I), is in no way inferior to their most beautiful discoveries in other branches of Mathematics. The existence of the gcd of two integers is proved right at the beginning of Book VII by the procedure known as the « Euclidean algorithm » \* ; it serves as the basis for all the subsequent developments (properties of prime numbers, existence and calculation of Icm, etc.), based on arguments not substantially different from those in Chap. VI, 9 1 above ; and the crowning glory is formed by the two remarkable theorems proving the existence of infinity of prime numbers (Book IX, Prop. 20) and giving a procedure for constructing even perfect numbers (cf. VII, p. 53, Ex. 24 ; this procedure in fact gives all the even perfect numbers, as would be proved by Euler). It is only the existence and uniqueness of prime factorisation that is not proved in a general manner ; however Euclid does prove explicitly that every integer is divisible by a prime number (Book VII, Prop. 31), as well as the following two propositions (Book IX, Prop. 13 and 14) :

« If as many numbers as we like are in a progression, beginning with 1, with constant ratio [i.e. geometric], and if the number following 1 is prime, then the

\* If $a_1$ and $a_2$ are two integers, such that $a_1 \geq a_2$, one defines $a_n$ (for $n \geq 3$) recursively as the remainder on dividing $a_{n-2}$ by $a_{n-1}$; if $m$ is the smallest index such that $a_m = 0$, then $a_1$, $a_2$, is the gcd of $a_1$ and $a_2$. This is the transposition into the integers of the method of successive subtractions (sometimes also called ἀνθυφαιρεσις) for finding a common measure for two lengths. The latter no doubt goes back to the Pythagoreans, and seems to have been the basis of a pre-Eudoxean theory of irrational numbers.

largest number will not be divisible by any number except those appearing in the progression » (in other words, a power $p^n$ of a prime number $p$ can be divisible only by powers of $p$ of exponent $\leq n$).

« If a number is the smallest which is divisible by [given] prime numbers, it will not be divisible by any other prime number with the exception of those initially [given as] dividing it » (in other words, a product of distinct primes $p_1, \ldots, p_k$ has no prime factor other than $p_1, \ldots, p_k$).

Thus it seems that if Euclid does not state the general theorem, it is only for want of an adequate terminology and notation for arbitrary powers of an integer.*

Although a careful study makes it seem likely that Euclid's text is made up of several successive layers, each corresponding to a stage in the development of Arithmetic **, it seems that this evolution took place entirely between the beginning of the 5th and the middle of the 4th centuries B.C., and one must admire the skill and the logical confidence which it represents: the next comparable progress made in Arithmetic would not occur for two millenia.

It was the so-called « indeterminate » or « Diophantine » problems which were at the root of subsequent developments in Number Theory. The term « Diophantine equations », as it is used today, is not entirely justified historically; it is generally understood to mean polynomial equations (or systems of equations) with integer coefficients, for which only integer solutions are sought: a problem which is usually impossible if the equations are « determinate », that is have only finitely many (real or complex) solutions, but which on the other hand often admits solutions when there are more unknowns than equations. Now, while Diophantus does indeed seem to have been the first to consider « indeterminate » problems, he looks for integer solutions only in exceptional cases, and is usually content to find a single solution in the rational numbers (II). That was a type of problem which he was usually able to solve by algebraic calculations, where the

\* In support of this thesis, one can also remark that the proof of the theorem on perfect numbers is basically just another special case of the unique prime factorisation theorem. Moreover, all the evidence shows that from this period onwards the factorisation of an explicit number into primes was well known and in current use; but no complete proof of the factorisation theorem is to be found before that given by Gauss at the beginning of the Disquisitiones ((VIII) t. I, p. 15).

** Cf. B. L. van der Waerden, Die Arithmetik der Pythagoreer, Math. Ann., vol. CXX (1947-49), p. 127. An example of a passage left over from a previous version is provided by Prop. 21 to 34 of Book IX, which deals with the most elementary properties of divisibility by 2, and no doubt goes back to a period when the general theory of prime numbers had not been developed. Moreover, it is known that the categories Even and Odd played a major rôle in the metaphysical speculations of the early Pythagoreans, to whom it is naturally tempting to attribute this segment.

arithmetical nature of the unknowns was not relevant *; also the theory of divisibility plays only a minor rôle (the word for prime number is used only once ((11), Book V, problem 9, vol. 1, pp. 334-335), and the notion of coprime numbers is invoked only in connection with the theorem which asserts that the quotient of two coprime numbers can be a square only if each of them is a square) **.

The study of integer solutions of indeterminate equations only begins properly with the Chinese and Hindu mathematicians of the early middle ages. The former seem to have been led to considerations of this nature by the practical problems of drawing up calendars (where the determination of the common periods of various cycles of astronomical phenomena constitutes precisely a linear « Diophantine » problem); at any rate they were responsible (certainly between the 4th and the 7th century A.D.) for a rule for solving simultaneous linear congruences (cf. VI, p. 33, Ex. 25). As for the Hindus, whose Mathematics flourished between the 5th and the 13th centuries, not only did they know how to deal methodically (by applying the Euclidean algorithm) with systems of linear Diophantine equations in arbitrarily many unknowns ***. but they were the first to attack and solve quadratic problems, among them certain special cases of « Fermat's equation » Nx^2 + 1 = y^2 ((111), vol. II, pp. 87-307).

This is not the place for us to pursue the history of the theory of Diophantine equations of degree > 1, which, by way of the work of Fermat, Euler, Lagrange and Gauss, was to lead to the theory of algebraic integers in the 19th century. As we have already noted (cf. Historical Note to Chap. II and 111), the study of linear systems, which no longer seemed to present problems worthy of interest, was rather neglected during this period: in particular there was no search for general existence conditions for an arbitrary system, nor for a description of the set of solutions. Nevertheless Hermite, in his Number-theoretical research in the 19th

\* While Diophantus' work on indeterminate problems is always reduced to problems in a single unknown, via a numerical choice of the other unknowns in such a way as to make a solution to his final equation possible, it seems that the main reason for using this method was his notation, which did not allow him to calculate with several unknowns at once; in any case he keeps track of the numerical substitutions he has made throughout the calculation, and modifies them later if need be, by writing down a compatibility condition for the substituted variables, and solving this auxiliary problem first. In other words, he handles these substituted numerical values like we handle parameters, so much so that what he actually does amounts to finding a rational parametric representation of a given algebraic variety, or of a subvariety of it (cf. (II bis)).

** Various indications, however, point to more advanced arithmetical knowledge in Diophantus: he knew for example that the equation x^2 - y^2 = n has no rational solutions if n is an integer of the form 4k + 3 (Book V, problem 9 and Book VI, problem 14 ((II), vol. 1, pp. 332-335 and p. 425; cf. also (II bis), pp. 105-110)).

*** Astronomical problems were also among those which led the Hindus to consider this type of equation (cf. (III), vol. II, pp. 100, 117 and 135).

century, was led to make use of several Lemmas on linear Diophantine equations, notably a « reduced form » for a linear transformation with integer coefficients ((XIII), pp. 164 and 265); finally, after Heger in 1858 had given the existence condition for a system whose rank is equal to the number of equations, H. J. Smith in 1861 defined the invariant factors of an integer matrix, and obtained the general theorem that such a matrix reduces to the « normal form » which we gave in VII, p. 22, Cor. 1 (XVII).

But meanwhile, following its introduction by Gauss (cf. Historical Notes to Chap. I, II and III), and the important part it played in the subsequent development of Number Theory, the notion of an abelian group was gradually being made precise. In his particularly deep study, presented in the Disquisitiones, of the finite abelian group of classes of quadratic forms of a given discriminant, Gauss soon realised that certain of these groups are not cyclic: « in this case, » he says, « one basis [that is to say, one generator] cannot suffice, it is necessary to take two or a greater number which, through multiplication and composition*, can produce all the classes » ((VIII), vol. I, pp. 374-375). It is not certain that Gauss intended these words to describe the decomposition of the group as a direct product of cyclic groups; nevertheless, in the same article of the Disquisitiones, he proves that there exists an element in the group whose order is the lcm of the orders of all the elements — in other words he obtains the existence of the largest invariant factor of the group ((VIII), vol. 1, p. 373); and on the other hand, the notion of direct product was known to him, for, in a manuscript dating from 1801, but not published during his lifetime, he sketches a general proof of the decomposition of a finite abelian group into a direct product of p-groups** ((VIII), vol. II, p. 226). In any case, in 1868 Schering, the editor of Gauss' collected works, inspired by these results (notably by this manuscript which he had just found), proved (still for the group of classes of quadratic forms) the general decomposition theorem (XVIII) by a method which, as repeated in abstract terms two years later by Kronecker (XX), is essentially that which we have used above (VII, p. 18, Th. 1). As for torsion-free abelian groups, we have already said (cf. Historical Note to Chap. II and III) how the theory of elliptic functions and abelian integrals, developed by Gauss, Abel and Jacobi, gradually led to attention being paid to their structure; the first and best known example of a decomposition of an infinite group into a direct sum of cyclic groups was given in 1846 by Dirichlet in his paper on the units of an algebraic number field (XI). But it was not until 1879 that the connection between the theory of finitely generated abelian groups and Smith's theorem was recognised and explicitly used by Frobenius and Stickelberger ((XXIII), § 10).

\* Gauss uses additive notation for the law of composition of classes, thus by « multiplication » he means the product of a class by an integer. ** Abel also proved this property en passant in his note on abelian equations ((IX), vol. I, pp. 494-497).

Around the same period the theory of similarity of matrices (with real or complex entries) was also reaching completion. The notion of an eigenvalue of a linear transformation appeared explicitly in the theory of systems of linear differential equations with constant coefficients, applied by Lagrange (VIa) to the theory of small oscillations and by Lagrange (VIb) and Laplace (VIIa) to the « secular » perturbations of the planets. It is implicit in many other problems which were also attacked around the middle of the 18th century, such as that of finding the axes of a conic or a quadric (first solved by Euler (Va)), or the study (also developed by Euler (Vb)) of the principal axes of inertia of a solid body (discovered by De Segner in 1755); we know now that it is also involved (in a more disguised form) in the beginnings of the theory of partial differential equations, in particular the equation of a vibrating string. But (leaving this last case aside) the relationship between these various problems was barely recognised before Cauchy (X). Moreover, as most of them involve the use of symmetric matrices, it was principally because of the latter that eigenvalues were initially studied; we will return to this point in more detail in the Historical Notes following the chapters of this treatise devoted to hermitian operators; let us simply note here that, as early as 1826, Cauchy proved the invariance of the eigenvalues of such matrices under similarity, and proved that they are real for a $3 \times 3$ symmetric matrix (Xa), a result which he generalised three years later (Xb) to arbitrary real symmetric matrices.* The general notion of a projection, introduced by Mobius in 1827, rapidly led to the problem of classifying such transformations (in 2 and 3 dimensions first of all), which is nothing other than the problem of classifying the corresponding matrices up to similarity; but for a long time this question was treated only by the « synthetic » methods which were in vogue in the mid 19th century, and its (in any case rather slow) progress does not seem to have affected the theory of eigenvalues in any way. The same is not true of another geometric question, the classification of « pencils » of conics or quadrics, which from the modern point of view amounts to the study of the elementary divisors of the matrix $U + \lambda V$, where $U$ and $V$ are two symmetric matrices; it was certainly in this spirit that Sylvester attacked this problem in 1851, carefully examining (in order to find « canonical forms » for the pencil under consideration) what happens to the minors of the matrix $U + \lambda V$ when a value for $\lambda$ is substituted which annihilates the determinant (XIV). The purely algebraic aspect of the theory of eigenvalues was progressing simultaneously; so it was that several authors (including Sylvester himself) proved around 1850 that the eigenvalues of $U^n$ are the n-th powers of the eigenvalues of $U$, while Cayley

\* An attempt to prove this result in the particular case of the « secular » perturbations of the planets had previously been made by Laplace (VIIb) in 1784. As for the third degree equation giving the axes of a real quadric, Euler had stated without proof that it has real roots, and an attempted proof by Lagrange in 1773 (VIe) was incomplete; this point was first proved rigourously by Hachette and Poisson in 1801 (Journal de l'École Polytechnique, cahier 11 (year X), pp. 170-172).

announced in 1858, in the paper in which he introduced matrix arithmetic (XVI), the « Cayley-Hamilton Theorem » for an arbitrary square matrix *, though he contented himself with a direct proof for 2 x 2 and 3 x 3 matrices. Finally Weierstrass in 1868, using Sylvester's methods, obtained « canonical forms » for a « pencil » $U + XV$ where this time $U$ and V are square matrices, not necessarily symmetric, subject only to the condition that $\det(U + XV)$ not be identically zero; he deduced from that the definition of the elementary divisors of an arbitrary square (complex) matrix, and proved that they characterise the latter up to similarity (XIX); these results, incidentally, were partially (and apparently independently) recovered by Jordan two years later ** (XXI). Here again it was Frobenius who showed in 1879 that Weierstrass' theorem can easily be deduced from Smith's theorem extended to polynomials ((XXII), § 13); his procedure is the basis of the proof of this theorem which we have given above (VII, p. 35).

We have just referred to the theory of divisibility for polynomials in one variable; the question of division of polynomials must naturally have arisen in the earliest days of algebra, as the inverse operation to multiplication (the latter being known even to Diophantus, at least for polynomials of low degrees); but one can imagine that it is barely possible to attack this problem in a general fashion before a coherent notation for the various powers of the variable has been established. In fact we find very few examples of the « Euclidean » division procedure for polynomials, as we know it, before the middle of the 16th century ***; and S. Stevin (essentially, using exponent notation) seems to have been the first to have had the idea of deducing the extension of the « Euclidean algorithm » to find the gcd of two polynomials ((IV, vol. I, pp. 54-56). Apart from that, the theory of divisibility had been restricted to the rational integers until the mid 18th century. It was Euler in 1770 who began a new chapter in Arithmetic by somewhat rashly extending the notion of divisibility to the integers of a quadratic extension: seeking to determine the divisors of a number of the form $x^2 + cy^2$ (x, y and c rational integers), he put

and, taking norms of each side, he had no hesitation in asserting that all the divisors of $x^2 + cy^2$ are obtained in this way as $p^2 + cq^2$ (Vc). In other words,

\* Hamilton, incidently, had proved this theorem for 3 x 3 matrices some years previously ((XV), pp. 566-567).

** Jordan did not mention the invariance of the normal form which he obtained. It is interesting to note as an aside that he dealt with the problem not for complex matrices, but for matrices over a finite field. On the other hand, let us point out that Grassmann had given a method for reducing a (complex) matrix to triangular form as early as 1862, and explicitly mentioned the connection between this reduction and the classification of projections (Ges. Math. Werke, vol. I2, Leipzig (Teubner), 1896, pp. 249-254).

*** Cf. for example H. Bosmans, Sur le « libro del Algebra » de Pedro Nuñez, Bibl. Math. (3), vol. VIII (1907-1908), pp. 154-169.

Euler argued as if the ring $\mathbf{Z}[\sqrt{-c}]$ were a principal ideal domain; a little later he used an analogous argument to apply the method of « infinite descent » to the equation $x^3 + y^3 = z^3$ (he reduced the problem to that of finding a cube root of $p^2 + 3q^2$, which he does by putting $p + q \sqrt{-3} = (r + s \sqrt{-3})^3$). But Lagrange proved as early as 1773 (VIc) that the divisors of numbers of the form $x^2 + cy^2$ are not all of this form, the first example of the fundamental difficulty which was to appear much more clearly in the studies of Gauss and his successors into divisibility in cyclotomic fields ; it is not possible, in general, to extend the essential properties of divisibility of rational integers, such as the existence of gcd's and the uniqueness of prime factorisation, directly to these fields. This is not the place to describe how Kummer, for cyclotomic fields (XII)*, and then Dedekind and Kronecker for arbitrary algebraic number fields, succeeded in overcoming this formidable obstacle by the invention of ideal theory, one of the most decisive advances of modern algebra. But Dedekind, ever curious about the foundations of various Mathematical theories, was not content with this success; and by analysing the mechanism of the divisibility relations he laid the foundations of the theory of lattice ordered groups, in a paper (unknown to his contemporaries, and lost in obscurity for 30 years) which is without doubt one of the earliest works of axiomatic Algebra (XXIV); modulo notation, his work was very close to the modern form of this theory, as we have presented it in Chap. VI, 91.

\* Gauss seems to have hoped at one time that the ring of integers in the field of n-th roots of unity would be a principal ideal domain; in a manuscript unpublished in his lifetime ((VIII), vol. II, pp. 387-397), he proves the existence of a Euclidean division process in the field of cube roots of unity, and gives some indications of an analogous process in the field of 5th roots of unity; he uses these results to prove by an « infinite descent » argument more correct than Euler’s that the equation $x^3 + y^3 = z'$ has no solution in the field of cube roots of unity, indicates that one can extend the method to the equation $x^5 + y^5 = z^5$, but stops short of the equation $x^7 + y^7 = z^7$, saying that here it is impossible to reject apriori the case where x, y and z are not divisible by 7.

** In his earliest work on « ideal numbers », Kummer explicitly indicated the possibility of applying his method, not only to cyclotomic fields, but also to quadratic fields, and hence of recovering Gauss' results on binary quadratic forms ((XII), pp. 324-325).

(Vd): for every polynomial $f$ with real coefficients, he attempted to prove the existence of a factorisation $f = f_1 f_2$ into two (non constant) polynomials with real coefficients, which would give him a proof of the « fundamental theorem » by induction on the degree off. It even suffices, as he noticed, to stop at the first odd degree factor, and consequently all the difficulties are reduced to the consideration of the case where the degree $n$ off is even. Euler then restricts himself to the case when the desired factors each have degree $n/2$, and he indicates that by a suitable process of elimination one can express the unknown coefficients of $f_1$ and $f_2$ as rational functions of a root of an equation with real coefficients, whose extreme terms have opposite signs and which consequently has at least one real root. But Euler's proof was only a sketch, and skipped over a number of essential points; it was not till 1772 that Lagrange succeeded in resolving the difficulties raised by this proof (VIId) by means of an extremely long and minute analysis, in which he demonstrated a remarkable virtuosity in the use of the « Galois methods » which he had recently created (cf. Historical Note to Chap. IV and V).

All the same, Lagrange, like Euler and all his contemporaries, had no hesitation in arguing formally within a « field of roots » of a polynomial (that is, in his language, to consider « imaginary roots » of this polynomial); the Mathematics of his period had provided no justification for this type of argument. Gauss, who was from the outset resolutely hostile to the unrestricted formalism of the 18th century, came out strongly against this abuse in his dissertation ((VIII), vol. III, p. 3). But it would have been unlike him not to have sensed that it was a case of a superficially faulty presentation of an argument which was intrinsically correct. We find him also, a few years later ((VIII), vol. III, p. 33; cf. also (VIIIbis)), taking up a simpler form of Euler's argument, which had been suggested in 1759 by Foncenex (who, however, had been unable to use it to any advantage), to obtain a new proof of the « fundamental theorem » in which he carefully avoids all mention of « imaginary » roots: the latter being replaced by skilful adjunction and specialisation of indeterminates. It is essentially this proof of Gauss that we have given in the text (VI, p. 26, Th. 3), with simplifications made possible by the use of algebraic extensions.

The rôle of Topology in the « fundamental theorem » was thus reduced to the single theorem that a polynomial with real coefficients cannot change sign in an interval without having a zero (Bolzano's Theorem for polynomials). This theorem is also at the root of all the criteria for separation of the real roots of a polynomial (with real coefficients), which was one of the favourite Algebraic topics in the 19th century *. In the course of this research it becomes obvious that it is the order structure of $\mathbf{R}$, much more than its topology, which plays the

\* On these topics, which we will not discuss in the present work, the reader may consult for example J.-A. Serret, Cours d'Algèbre supérieure, 3rd ed., Paris (Gauthier-Villars), 1866, or B. L. van der Waerden, Moderne Algebra, vol. I (1st ed.), Berlin (Springer) 1930, pp. 223-235.

essential rôle *; for example Bolzano's Theorem for polynomials remains true in the field of all real algebraic numbers. This train of ideas resulted in the abstract theory of ordered fields, created by E. Artin and O. Schreier (XXV); one of its most remarkable results is certainly the discovery that the existence of an order relation on a field is related to purely algebraic properties of the field. This is the theory presented in § 2 of Chap. VI.

\* The tendency to attach overwhelming importance to the order structure of the real numbers can also be seen in their definition by means of Dedekind « cuts », which is basically a process which can be applied to all ordered sets (cf. VI, p. 35, Ex. 30 onwards).

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