5. Polynomials and commutative fields.

The theory of commutative fields — and the theory of polynomials, which is closely linked to it — derive directly from what constituted, until the middle of the XIXth century the principal object of classical Algebra: the solution of algebraic equations, and geometric construction problems which are equivalent to that.

As soon as the solution of an algebraic equation of degree $> 1$ is sought, one finds oneself up against calculation difficulties of a totally new kind, the determination of an unknown being impossible to accomplish by means of "rational" calculations starting from given quantities. This difficulty must have been seen at a very early stage: and it must be counted as one of the most important contributions of the Babylonians that they knew how to reduce the solution of quadratic and biquadratic equations to a single new algebraic operation, the extraction of square roots (as is proved by the numerous numerical equations that are found solved in this way in the texts that have reached us ([232], pp. 183-193)). As far as formal calculation is concerned, Antiquity will never reach further than this point in the problem of solving algebraic equations; the Greeks of the classical era limit themselves indeed to finding again the Babylonian formulae in geometrical terms, and their use in an algebraic form is not attested to before Hero (100 after J. C.) and Diophantus.

It is in a totally different direction that the Greeks make decisive progress. We have poor information on the way in which the Babylonians conceived of and calculated the square roots of non square integers;$^1$ in the rare texts which have reached us on this question, they seem to be satisfied with fairly crude approximation methods ([232], pp. 33-38). The Pythagorean school which had determined with rigour the concept of commensurable quantities, and attached to it a quasi religious character, could not hold this point of view; and it is possible that it was the failure of repeated attempts to express $\sqrt{2}$ as a rational that led them finally to prove that this number is irrational.$^2$

$^1$ In all the examples of quadratic and biquadratic equations in the Babylonian texts, the given quantities are always chosen so that the radicals always apply to squares.

$^2$ A recent author has made the ingenious remark that the construction of the regular starred pentagon, known to the Pythagoreans (of whom it was one of the mystic symbols) led immediately to a proof of the irrationality of $\sqrt{5}$, and has put

We will say elsewhere (cf. p. 148) how this discovery, which marks a major turning point in the history of Mathematics, reacted profoundly on the concept of "number" among the Greeks, and led them to create an algebra of exclusively geometric character, so as to find in it a way of representing (or perhaps a proof of "existence" for) the incommensurable ratios, that they refused to consider as numbers. Most often, an algebraic problem is reduced by them to the intersection of two auxiliary plane curves, suitably chosen, or to several successive determinations of such intersections. Late traditions with little authority make the introduction of a first classification of these constructions, destined for a long and brilliant career, go back to Plato: for reasons that are more philosophical than mathematical, it appears, he would have put aside the constructions "by ruler and compass" so called, that is to say those where there are no auxiliary curves except for straight lines and circles. In any case, Euclid, in his Elements [107], limits himself exclusively to treating problems soluble in this way (without however characterising them by a particular name); a circumstance which doubtless would contribute not a little to fastening the attention of mathematicians of the following centuries on these problems. But we know today that algebraic equations that can be solved "by ruler and compass" are of a very special type; in particular, an irreducible equation of the 3rd degree (over the field of rationals) can not be solved in this way, and the Greeks had met very early on problems of the 3rd degree which have remained famous, such as the duplication of the cube (solution of $x^3 = 2$) and the trisection of an angle; the quadrature of the circle put them on the other hand in presence of a transcendental problem. We see forth the hypothesis (which unfortunately is not supported by any text) that it is in this way that the Pythagoreans were to discover irrational numbers [323].

3 Linked to this principle, the classification of plane curves in "plane localities" (straight line and circle), "solid localities" (the conics, obtained by a plane section of a solid body, the cone) is also attributed to Plato, all the other curves being grouped under the name of "τόποι γραμμικοί". It is odd to see the influence of this classification exerting itself on Descartes, who, in his Geometry, ranks in the same "genre" the equations of degree $2n - 1$ and of degree $2n$, no doubt because those of degree 1 or 2 are solved by the intersection of "plane localities" and those of degree 3 or 4 by intersections of "solid localities" ([85 a], v. VI, pp. 392-393).

4 The determination of the points of intersection of a straight line and a circle (or of two circles) is equivalent to the solution of an equation of the second degree of which the coefficients are rational functions of the coefficients of the equations of the straight line and the circle (or of the two circles) under consideration. It is easily concluded that the co-ordinates of a point constructed "by ruler and compass" starting from given points lies in an extension $L$ of the field $\mathbf{Q}$ of rational numbers, obtained as follows: if $K$ is the field obtained by adjoining to $\mathbf{Q}$ the co-ordinates of the given points, there exists an increasing sequence $(L_i)_{0 \leq i \leq n}$ of intermediate fields between $K$ and $L$, satisfying the conditions $K = L_0, L = L_n, [L_i : L_{i-1}] = 2$ for $1 \leq i \leq n$. By induction on $n$, it is deduced that the degree over $K$ of the Galois extension $N$ generated by $L$ is a power of 2; conversely, one can prove that, if this condition is fulfilled, there exists a sequence $(L_i)$ of fields intermediate between $K$ and $L$ having the previous properties, and it follows that the problem as stated is soluble by ruler and compass (cf. [312], pp. 351-366).

them, in order to solve these problems, introducing numerous algebraic (conics, cissoid of Diocles, conchoid of Nicomedes) or transcendental (quadratic of Dinostratus, spiral of Archimedes) curves; which could not have failed to lead them to an autonomous study of these various curves, thus preparing the way for future developments in analytic Geometry, in algebraic Geometry and in infinitesimal Calculus. But these methods do not help to make any progress in the solution of algebraic equations,$^5$ and the only work of Antiquity which made a worthy contribution to this question and exerted a lasting influence on the algebraists of the Middle Ages and the Renaissance is Book X of the Elements of Euclid [107]; in this Book (to which some historians would make the principal results of Theetetus go back), he considers the expressions obtained by the combination of several radicals, such as $\sqrt{\sqrt{a} \pm \sqrt{b}}$ ($a$ and $b$ rational), gives conditions under which these expressions are irrational, classes them into numerous categories (which he shows are distinct), and studies the algebraic relations between these various irrationals, such as those which we would write today as

the whole expressed in the usual geometric language of the Elements, which makes the exposition particularly heavy and awkward.

After the decline of classical Greek mathematics, the concepts concerning algebraic equations are modified. There is no doubt that, during the whole of the classical period, the Greeks were in possession of methods of indefinite approximation for square roots, about which we are unhappily badly informed.$^6$ With the Hindus, then the Arabs and their western emulators in the Middle Ages, the extraction of roots of all orders becomes an operation that tends to be considered as fundamental on the same basis as the rational operations of algebra, and to be denoted, like these latter, by symbols which

$^5$ Because of the lack of a manageable algebraic calculus, no trace is found among the Greeks of an attempt to classify those problems that they did not know how to solve by ruler and compass: the Arabs are the first to reduce many problems of this type (for example the construction of regular polygons with 7 and 9 sides) to equations of the 3rd degree.

$^6$ For example, the method of Archimedes for the approximate calculation of the number $\pi$ requires the knowledge of several square roots, to a fairly good approximation, but we are ignorant of the procedure used by Archimedes to obtain these values. The method of approximation for $\sqrt{2}$ which supplies the expansion of this number as a "continued fraction" is known (in a geometric form) from a text of Theon of Smyrna (IIInd century after J. C.), but goes back perhaps to the first Pythagoreans. As for the method of indefinite approximation of square roots still in use in our day in elementary teaching, it is not attested to before Theon of Alexandria (IVth century after J. C.), even though it was no doubt known already to Ptolemy. Note finally that one finds in Hero (about 100 after J. C.) an approximate calculation for a cube root [103].

are more and more manageable in calculations.$^7$ The theory of the equation of the second degree, which is perfected during the whole of the Middle Ages (number of roots, negative roots, cases of impossibility, double roots), and that of biquadratic equations, give models for formulae for the solution of equations "by radicals", from which algebraists are going to try for centuries to copy analogous formulae for the solution of equations of higher degree, and in the first instance for equations of the 3rd degree. Leonardo of Pisa, the principal introducer of Arab science to the West in the XIIIth century, recognises in any case that the irrationals classified by Euclid in his Xth book can not be of use to this end (a new proof of impossibility, in a theory that contains so many), and we see him already trying to do similar calculations with cube roots, obtaining relations such as

analogous to Euclid's formulae for square roots (and of which besides previous examples are found among the Arabs). But three centuries of unfruitful efforts will still go by before Scipio del Ferro, at the beginning of the XVIth century, finally succeeds, for the equation $x^3 + ax = b$, in obtaining the solution formula$^8$

We cannot describe here the picturesque side of this sensational discovery — the quarrels that it provoked between Tartaglia on the one hand, Cardan and his school on the other — nor the figures, often appealing, of the scholars who were the protagonists. But we must note the decisive progress that ensued for the theory of equations, in the hands of Cardan and his pupils. Cardan, who had less repugnance than most of his contemporaries at using negative numbers, observes thus that equations of the third degree may have three roots, and the biquadratic equations four ([50], v. IV, p. 259), and he remarks that the sum of the three roots of $x^3 + bx = ax^2 + c$ (an equation in which he knows how to eliminate the term in $x^2$) is always equal to $a$ (ibid.). No doubt guided by this relation and the intuition of his general character, he has the initial idea of the notion of multiplicity of a root; above all he is emboldened (not without oratorical preparation) to calculate formally with expressions containing square roots of negative numbers. It is likely that he was brought to this by the fact that such expressions arise naturally in the

$^7$ The irrationality of $\sqrt[n]{a}$, when $a$ is an integer which is not an exact $n$th power, is not mentioned or proved before Stifel (XVIth century); the proof of this latter ([298], p. 103) is in any case copied from that of Euclid for $n = 2$, and it is fairly unlikely that this easy generalisation had not been noticed before.

$^8$ If we put $x = y + z$ with the condition $yz = -a/3$ we obtain $y^3 + z^3 = b$ and $y^3 z^3 = -(a/3)^3$, from which we get $y^3$ and $z^3$.

use of the formula (1) when $(\frac{b}{2})^2 + (\frac{a}{3})^3 < 0$ (a case — called “irreducible” — where Cardan had recognised the existence of three real roots); it is what appears in any case with clarity in his disciple R. Bombelli, who, in his Algebra ([28 a], p. 293) proves the relation

and takes care to give explicitly the rules for the calculation of complex numbers in a form which is very close to modern expositions.$^9$

Finally, already in 1545, another pupil of Cardan, L. Ferrari, succeeds in solving the general equation of the 4th degree, with the help of an auxiliary equation of the 3rd degree.$^{10}$

After such a rapid advance, the following period, until the middle of the XVIIIth century, does little more than develop the new ideas introduced by the Italian school. Thanks to the essential progress that it brings to algebraic notation, Viète can express in a general manner the relations between the coefficients and roots of an algebraic equation, at least when the roots are all positive$^{11}$ ([319], p. 158). Bolder, A. Girard [129] does not hesitate to affirm (of course without proof) that an equation of degree $n$ has exactly $n$ roots, on condition that “impossible roots” are counted, each with its degree of multiplicity, and that these roots satisfy the relations given by Viète; he obtains also, for the first time, an expression for the sums of powers of roots up to exponent 4.

But the spirit of the XVIIth century is turned towards other directions and it is only as an indirect consequence that Algebra profits somewhat from the new discoveries of analytic Geometry and the infinitesimal Calculus. So, to the method of Descartes for obtaining the tangents to algebraic curves (cf. p. 178) is linked the criterion for the multiplicity of a root of an algebraic equation, which is stated by his disciple Hudde ([85 b], v. I, pp. 433 and 507-509). It is no doubt also to the influence of Descartes that one must go back

$^9$ Bombelli ([28 a], pp. 169 and 190) considers complex numbers as “linear combinations” with positive coefficients, of four basis elements: “piu” (+1), “meno” (-1), “piu de meno” (+i) and “meno de meno” (-i); he states notably as an axiom that “piu” and “piu de meno” can not be added, first appearance of the notion of linear independence.

$^{10}$ The equation being reduced to the form $x^4 = ax^2 + bx + c$, a number $z$ is determined such that the second member of the equation

$^{11}$ Viète, a passionate admirer of the Ancients, abstains systematically from introducing negative numbers into his arguments; he is none the less capable, on occasions, of expressing in his language relations between coefficients and roots when some of these latter are negative; for example, if the equation $x^3 + b = ax$ has two positive roots $x_1, x_2 (a > 0, b > 0)$, Viète shows that $x_1^2 + x_2^2 + x_1 x_2 = a$ and $x_1 x_2 (x_1 + x_2) = b$ ([319], p. 106).

for the distinction between algebraic functions and transcendental functions, parallel to that which he introduces in his Géométrie between "geometric" and "mechanical" curves (cf. pp. 176 and 193.). In any case, this distinction is perfectly clear in J. Gregory who, in 1667, tries even to prove that the area of a circular sector can not be an algebraic function of the chord and radius ([136 a]; cf. [155]). The expression "transcendental" is from Leibniz, who does not cease to be interested in these questions of classification for the whole of his career, and who, in about 1682, discovers a simple proof of the result sought by Gregory, by proving that $\sin x$ can not be an algebraic function of $x$ ([198 a], v. V, pp. 97-98).12 With his friend Tschirnhaus, he is besides one of the only mathematicians of his time who is still interested in the problem of the solution "by radicals" of algebraic equations. In his early stages, we see him studying the "irreducible case" of the equation of the 3rd degree, and convincing himself (in fact without sufficient proof) that it is impossible in this case to get rid of imaginary quantities from the formulae of the solution ([198 d], pp. 547-564). At about the same time, he attacks also without success the solution by radicals of the equation of the 5th degree; and when later, Tschirnhaus pretends to have solved the problem by making all the terms of the equation disappear except for the two extremes by means of a transformation of the form $y = P(x)$, where $P$ is a polynomial of the 4th degree, suitably chosen, Leibniz realises straightaway that the equations which determine the coefficients of $P(x)$ are of degree $> 5$, and judges that the method is destined for failure ([198 d], pp. 402-403).

It appears that it is the needs of the new Analysis that had little by little revived interest in algebra. The integration of rational functions, accomplished by Leibniz and Johann Bernoulli, and the question of imaginary logarithms which is closely linked to it, are the occasion for deepening the calculus of imaginary numbers, and for answering the question of the decomposition of a polynomial into factors of the first degree ("fundamental theorem of algebra").13 From the beginning of the XVIIIth century, the solution of the binomial equation $x^n - 1 = 0$ is reduced by Cotes and de Moivre to

12 The definition that Leibniz gives of "transcendental quantities" ([198 a], v. V, p. 228; see also ibid., p. 120) seems to apply more to functions than to numbers (in modern language, what he does reduces to defining transcendental elements over the field obtained by adjoining to the field of rational numbers the given numbers from the problem); it is however likely that he had a fairly clear notion of transcendental numbers (even though these latter do not appear to have been defined in a precise way before the end of the XVIIIth century); in any case, he observes explicitly that a transcendental function can take rational values for rational values of the variable, and it follows that his proof of the transcendence of $\sin x$ is not sufficient to prove that $\pi$ is irrational ([198 a] v. V, pp. 97 and 124-126). 13 One can take full cognisance of the rudimentary state in which the calculus of complex numbers was still to be found at this time when Leibniz (nevertheless one of the most proficient at these techniques, amongst the mathematicians of his time) is seen expressing himself as if it were not possible to decompose $x^4 + 1$ into two real factors of the second degree ([198 a], v. V, pp. 359-360).

the division of the circle into $n$ equal parts; in order to obtain the expression of its roots "by radicals", it is sufficient therefore to know how to do it when $n$ is an odd prime, and de Moivre remarks that the substitution $y = x + \frac{1}{x}$ reduces the problem then to the solution "by radicals" of an equation of degree $(n - 1)/2$. As far as the "fundamental theorem" is concerned, after the repeated setbacks in the general solution "by radicals" (including several attempts by Euler ([108 a], (1), v. VI, pp. 1-19 and 170-196)), proofs begin to be sought $a$ priori, not using explicit formulae for the solution. Without going into the detail of the proposed methods (which would end up with the proofs of Lagrange and Gauss; cf. p. 91 and 161), it is fitting to note here the point of view from which the problem is seen in the middle of the XVIIIth century: it is assumed (without any justification other than a vague feeling of generality, coming no doubt, as with A. Girard, from the existence of relations between the coefficients and roots) that an equation of degree $n$ always has $n$ "ideal" roots, with which one can calculate as with numbers, without knowing if they are numbers (real or complex); and what it is necessary to prove is (by using if necessary calculations on ideal roots) that at least one of these roots is an ordinary complex number.$^{14}$

In this defective form, it is recognised that here is the first embryo of the general idea of a "formal adjunction" which, in spite of the objections of Gauss ([124 a], v. III, p. 1), would become the basis of the modern theory of commutative fields.

With the fundamental memoirs of Lagrange ([191], v. III, pp. 205-421) and of Vandermonde [315], the year 1770 sees a new and decisive period in the history of the theory of algebraic equations opening up. To the empiricism of the more or less fortunate attempts at solution formulae, which until then had reigned without dispute, a systematic analysis of the problems as stated and of the methods susceptible to solve them is going to follow, an analysis which in sixty years will lead to the definitive results of Galois. Lagrange and Vandermonde both start from the ambiguity which multiple determinations of radicals introduce in the formulae for the solution of equations of degree $\leq 4$; this fact had attracted the attention of Euler ([108 a], (1), v. VI, pp. 1-19) who had showed among other things how, in the formula of del Ferro, one must associate the determinations of the radicals which appear there so as to obtain 3 roots and not 9.$^{15}$ Lagrange remarks that each of the cubic radicals in the formula of del Ferro can be written in the form $\frac{1}{3}(x_1 + \omega x_2 + \omega^2 x_3)$, where $\omega$ is a cube root of one, $x_1, x_2, x_3$ are the three roots of the equation in question, taken in a certain order, and he makes the fundamental observation that the function $(x_1 + \omega x_2 + \omega^2 x_3)$ of the three roots can only take two distinct values for all permutations of the three roots, which explains a priori the success of the methods for the solution of this equation. A similar analysis

$^{14}$ It must be noted that what the mathematicians of the XVIIIth century call "imaginary roots" are often only the "ideal" roots above, and they seek to prove that these roots are of the form $a + b \sqrt{-1}$ (see for example [191], v. III, p. 479).

$^{15}$ See footnote of p. 72: one must have $yz = -a/3$.

of the solution methods for the equation of the 4th degree leads him to the function $x_1 x_2 + x_3 x_4$ of the four roots, which only takes three distinct values for all permutations of the roots, and is as a consequence a root of an equation of the third degree with coefficients that are rational functions of those of the given equation;$^{16}$ these facts constitute, says Lagrange, "the true principles, and, so to say, the metaphysic$^{17}$ of the solution of equations of the 3rd and of the 4th degree" ([191], v. III, p. 357). Relying on these examples, he proposes studying in general, for an equation of degree $n$, the number $\nu$ of values$^{18}$ that a rational function $V$ of the $n$ roots can take when they are arbitrarily permuted; in reality he inaugurates thus (with this terminology still tightly adapted to the theory of equations) the theory of groups and that of fields, of which he obtains already several fundamental results by use of the same principles as those that are used today. For example, he shows that the number $\nu$ is a divisor of $n!$ by means of the argument that is used today for proving that the order of a subgroup of a finite group divides the order of the group. More remarkable still is the theorem where he shows that, if $V_1$ and $V_2$ are two rational functions of the roots such that $V_1$ and $V_2$ are invariant under the same permutations, then each of them is a rational function of the other and the coefficients of the equation (a particular case of the theorem of Galois characterising a subextension of a Galois extension as the field of invariants of its Galois group): "This problem", he says, "appears to me one of the most important in the theory of equations, and the general solution that we will give for it will be the means of throwing new light on this part of Algebra" ([191], v. III, p. 374).

All these researches are naturally, in the mind of Lagrange, preliminaries to the analysis of possible methods for the solution of algebraic equations by a successive reduction to equations of lesser degree, such a method being linked, as he shows, to the formation of rational functions of the roots taking less than $n$ values under permutations of the roots. Guided no doubt by his results on the equation of the 3rd degree, he introduces in general the "Lagrange resolvents" $y_k = \sum_{h=1}^n \omega_k^h x_h$, where $\omega_k$ is an $n$th root of unity ($1 \leq k \leq n$), shows clearly how the knowledge of these $n$ numbers results in that of the roots $x_k$, and looks in general for the degree of the equation which is satisfied by the $y_k$; he shows for example that, if $n$ is prime, the $y_k$ are

$^{16}$ Waring also makes this observation in his Meditationes algebraicae, appearing in that same year 1770, but he is far from drawing out the same consequences as Lagrange. $^{17}$ With this word, which comes back so often from the pen of authors of the XVIIIth century, it is permissible to see a first intuition (still quite vague) of the modern conception of structure. $^{18}$ Lagrange already makes a distinction between the various rational fractions that are obtained starting from $V$ by permutations of the indeterminates $x_i (1 \leq i \leq n)$, and the various values that these fractions take when the $x_i$ are the roots of an algebraic equation with given numerical coefficients; but there still remains in his exposition a certain vagueness about this subject, and it is only with Galois that the distinction becomes much clearer.

roots of an equation of degree $n - 1$, whose coefficients are rational functions of a root of an equation of degree $(n - 2)!$ with coefficients that are expressed rationally in terms of the coefficients of the given equation. "Here are, if I am not mistaken", he concludes, "the real principles for the solution of equations, and the most correct analysis for getting there; everything reduces, as can be seen, to a kind of calculation of combinations, by which one finds a priori the results which should be expected" ([191], v. III, p. 403).

As for the memoir of Vandermonde, independent of that of Lagrange, it meets this latter at many points, particularly so far as the idea of seeking rational functions of the roots taking as few distinct values as possible under permutations of the roots is concerned,$^{19}$ and the study of the "Lagrange resolvents" that he also introduces for this purpose. His work is far from having the clarity and generality of that of Lagrange; on one point however it goes clearly further, in applying the same ideas to the equation for the division of the circle $x^n - 1 = 0$ for $n$ an odd prime. While Lagrange is content to remind readers that this equation reduces to an equation of degree $m = (n - 1)/2$, with rational coefficients, without seeking to solve it when $n \geq 11$, Vandermonde states that the $m$th powers of the Lagrange resolvents of this equation are rational, because of the relations between the various roots of $x^n - 1 = 0$; but he limits himself to checking the soundness of this assertion in the case $n = 11$, without justifying it in a general way.

It is only 30 years later that the result announced by Vandermonde was fully proved by C. F. Gauss.$^{20}$ His decisive results on the equation $x^n - 1 = 0$ ($n$ an odd prime) fit into the general programme of his memorable research on arithmetic ([124 a], v. I, pp. 413 ff.), and illustrate in a very special way his mastery in the handling of what we would now call the theory of cyclic groups. After having proved that the polynomial $\Phi_n(x) = (x^n - 1)/(x - 1)$ is irreducible for $n$ an odd prime,$^{21}$ he has the idea of writing its $n - 1$ roots in the form $\zeta^{g^k} = \zeta_k (0 \leq k \leq n - 2)$, where $g$ is a primitive root of the congruence $z^{n-1} \equiv 1 \pmod{n}$) (which amounts, in modern language, to bringing into the open the fact that the group $\Gamma$ of the equation $\Phi_n(x) = 0$ is cyclic). To every divisor $e$ of $n - 1$, he puts in correspondence the $f = (n - 1)/e$ "periods" $\eta_\nu = \xi_\nu + \xi_{\nu + f} + \xi_{\nu + 2f} + \ldots + \xi_{\nu + (e-1)f} (1 \leq \nu \leq f)$ and shows

$^{19}$ In this research (that he only develops indeed for the equation of the 5th degree) appears for the first time the notion of imprimitivity ([315], pp. 390-391). It is besides naturally tempting to bring together the methods of Lagrange and Vandermonde and their contemporaneous work on determinants, which must have made the idea of permutation and all that goes with it familiar to them

$^{20}$ Gauss does not quote Vandermonde in his Disquisitiones, but it is likely that he would have read the memoir of this latter (cf. [124 a], v. X_2, Abh. 4., p. 58).

$^{21}$ The notion of irreducible polynomial (with rational coefficients) goes back to the XVIIth century, and Newton and Leibniz had already set out procedures allowing (at least theoretically) the determination of the irreducible factors of a polynomial with explicit rational coefficients ([198 a], v. IV, pp. 329 and 355); but the proof of Gauss is the first proof of irreducibility which is applicable to a whole set of polynomials of arbitrarily high degree.

substantially that the linear combinations with rational coefficients of the $\eta_\nu$ make up a field, generated by any one of the $f$ periods $\eta_\nu$ and of degree $f$ over the field of rational numbers (this field corresponds naturally to the subgroup of $\Gamma$ of order $e$). We cannot go into the details of his analysis here, and of the important arithmetical consequences that follow from it; let us notice only that it gives him in particular the famous theorem on the possibility of constructing "by ruler and compass" the polygons having a number of sides equal to a prime number of the form $2^{2^k} + 1$.$^{22}$ As for the solution by radicals of the equation $\Phi_n(x) = 0$, it follows easily from the theory of periods, applied to the $f$th power of a Lagrange resolvant $\sum_{\nu=0}^{f-1} \omega^\nu \eta_\nu$ (where $\omega^f = 1$).$^{23}$

It is directly to Lagrange that the research of his compatriot Ruffini [265] is linked, contemporaneous with the Disquisitiones; taking up again the question at the point at which Lagrange had left it, he proposes as goal the proof of the impossibility of the solution "by radicals" of the "general" equation$^{24}$ of the 5th degree. The proof of Ruffini, prolix and obscure, remains incomplete although reworked several times; but it is already very close to the proof (correct in principle) that Abel would obtain later.$^{25}$ His main interest resides especially in the introduction of calculations with substitutions and the first notions of the theory of groups, that Ruffini develops to show that there does not exist a function of the 5 roots of the equation taking more than 2 and less than 5 values when the roots are arbitrarily permuted.

We have already said (see p. 51) how this first sketch of the theory of groups of permutations was developed and systematised by Cauchy a few years later. But, if the notions necessary for the development of the ideas of Lagrange are thus clarified little by little as far as substitutions are concerned, it was still necessary to state in as clear a way the first principles of the theory of fields. That was what was missing for Ruffini, and it is that which Abel

$^{22}$ Gauss affirms explicitly having a proof of the fact that this case is the only one where it is possible to construct by ruler and compass a polygon having an odd prime number of sides ([124 a], v. I, p. 462); but this proof was never published and has not been found in his papers. $^{23}$ In actual fact, if we only wish to prove that the equation is soluble by radicals, we can limit ourselves to taking $e = 1$, and arguing by induction on $n$. $^{24}$ The mathematicians of the XIXth century understood by that, in substance, an equation of which the coefficients are indeterminates over the field of rationals. But the modern notion of indeterminate does not emerge much before the last years of the XIXth century; until then, by "polynomial" or "rational function" what is meant is a function of complex variables. A "general" algebraic equation is conceived as an equation of which the coefficients are independent complex variables, and of which the roots are "algebraic functions" of these variables — notion truth to tell completely devoid of a precise meaning if one gives to the word "function" its actual meaning. Of course, the arguments about these "algebraic functions" are in general intrinsically correct, as can be verified by translating them into modern algebraic language. $^{25}$ The papers of Ruffini have been very carefully analysed in [43].

and Galois are going to do, in the last phase of the problem of the solution of algebraic equations.

During the whole of his short life, Abel does not fail to be concerned with this problem. While almost still a child, he believed that he had obtained a formula for the solution by radicals of the general equation of the 5th degree. Having realised later his mistake, he did not rest until he had succeeded in proving that such a formula did not exist ([1], v. I, p. 66). But he does not stop there. While his emulator Jacobi develops the theory of elliptic functions as an analyst, it is the algebraic point of view that dominates the work of Abel on this question, centred on the theory of the equations of division of elliptic functions ([1], v. I, pp. 265, 377 and passim). He obtains thus new types of equation soluble by radicals by a method copied from that of Gauss for the solution of equations of division of the circle ([1], v. I, pp. 310 and 358);26 a result from which he rises to the conception of “abelian” equations, for which he proves, in a famous memoir, the solubility by radicals ([1], v. I, p. 478); it is on this occasion that he defines in a precise way the notion of a polynomial irreducible over a given field (the field generated by the coefficients of the equation that he is studying).27 Finally death laid him low in 1829, just as he attacks the general problem of the characterisation of all the equations soluble by radicals, and having just communicated to Crelle and Legendre results already very near to those of Galois ([1], v. II, pp. 219-243, 269-270 and 279).

It was to him that was reserved, three years later, the crowning of the structure [123]. Like Abel, but in a still clearer way, he starts by defining (up to terminology) the membership by a number of a field generated by given numbers, the notion of adjunction, and irreducible polynomials over a given field. Given an equation $F(x) = 0$, without multiple roots, with coefficients in a given field $K$, he shows successively that “one can always construct a function $V$ of the roots, such that none of the values that are obtained by permuting in this function the roots in all possible ways, are equal”, that this function “enjoys the property that all the roots of the proposed equation can be expressed rationally as a function of $V$”, and that, $V, V', V''$, ... being the roots of the irreducible equation which $V$ satisfies, “if $a = f(V)$ is a root of

26 Gauss had already indicated, in the Disquisitiones, the possibility of generalising his methods to the equations of division of the lemniscate ([124 a], v. I, p. 413), and developed, in notes only published in our day, the particular case of division by 5 ([124 a], v. X, pp. 161-162 and 517). As with so many other brief and enigmatic indications that Gauss enjoyed sowing in his writings, the sentence of the Disquisitiones struck vividly at the mind of his contemporaries; and we know that it contributed not a little to inciting Abel and Jacobi in their researches on this question. 27 Even the notion of a field (as, more generally, that of set) is more or less a stranger to mathematical thinking before Cantor and Dedekind. Abel and Galois define the elements of their “base field” as being all those that can be expressed rationally as a function of the given quantities, without thinking of considering explicitly the set that these elements form.

the proposed equation, $f(V')$ as well will be a root of the proposed equation" ([123], pp. 47-51); in modern language, he thus proves that $V$, as well as any one of its conjugates over $K$, generates the field $N$ of the roots of $F$. He then defines the group $\Gamma$ of $F$ as the set of permutations of the roots $x_i$ that are obtained by substituting for $V$, in the rational expression for each of the $x_i$ as a function of $V$, any one of the conjugates of $V$; and he obtains straightaway the fundamental characterisation of the elements of $K$ by the property of being invariant under every permutation of $\Gamma$ ([123], p. 51). He proves next that, if $N$ contains the field of roots $L$ of another polynomial, the group of $N$ over $L$ is a normal subgroup of $\Gamma$ (a notion that he introduces at this point) ([123], p. 175). From there he deduces finally the criterion of solubility of an equation by radicals, by means of an argument of which this is the essential part: assuming that the base field $K$ contains all the roots of unity, there must exist by hypothesis an increasing sequence $(K_i)_{0 \leq i \leq m}$ of fields intermediate between $K$ and $N$, with $K_0 = K, K_m = N$, the field $K_{i+1}$ being obtained by adjoining to $K_i$ all the roots of a binomial equation $x^{n_i} - a_i = 0$ (with $a_i \in K_i$). There exists therefore in $\Gamma$ a decreasing sequence $(\Gamma_i)$ of subgroups such that $\Gamma_0 = \Gamma, \Gamma_m = e$ (neutral element), $\Gamma_{i+1}$ being normal in $\Gamma_i$ and the quotient group $\Gamma_i / \Gamma_{i+1}$ being cyclic (a case in which one says that the group $\Gamma$ is soluble). Conversely, if this is so, the use of a Lagrange resolvant shows that $K_{i+1}$ is obtained by adjoining to $K_i$ all the roots of a binomial equation, and hence the equation $F(x) = 0$ is soluble by radicals.$^{28}$ The impossibility of the solution by radicals of the "general" equation of degree $n > 4$ is then a consequence of the fact that the group $\Gamma$ of this equation, isomorphic to the symmetric group $S_n$ is not soluble.

Starting with the middle of the XIXth century, the algebraists, as we have already remarked (cf. p. 53), enlarge considerably the field of their investigations, until then almost entirely confined to the study of equations. In the light of the discoveries of Galois, it appears that the problem of the solution "by radicals" is only a particular case, fairly artificial, of the general problem of the classification of irrationals. It is this latter that, during the whole of the end of the XIXth century, is going to be attacked on several sides, and many disparate results will accumulate little by little, preparing the way for the synthesis of Steinitz.

$^{28}$ If $K$ does not contain all the roots of unity, and if $E$ is the field obtained by adjoining to $K$ all these roots, $E \cap N$ is an abelian extension of $K$; from which one deduces easily (by using the structure of finite abelian groups) that, in order for the group of $N$ over $K$ to be soluble, it is necessary and sufficient that the group of $E(N)$ over $E$ should be so. Taking into account the fact that the roots of unity are expressible "by radicals", one sees that the criterion of Galois is independent of all hypotheses about the field of numbers $K$ (and is valid more generally for all fields of characteristic 0). In fact, Galois does not make any simplifying hypothesis about $K$, and argues by induction on the order of the radicals which are successively adjoined to $K$ ([123], pp. 60-61).

First as far as algebraic irrationals are concerned, a fundamental principle of the classification was supplied by Galois theory, reducing the study of an algebraic equation to that of its group. In fact, it is mainly the theory of permutation groups, of which we do not have to talk here (cf. p. 53), that, in pure Algebra, is the principal object of research at that time. Other progress in the theory of algebraic fields comes from the development, at the same time, of Number Theory and of Algebraic Geometry. This progress is concerned mainly with the way of expounding the theory, and is, for the most part, due to Dedekind [79], who introduces the notions of field and ring,$^{29}$ and (linked to his research on hypercomplex systems) develops systematically the linear aspect of extension theory ([79], v. III; pp. 33 ff.). It is also he who considers the Galois group as made up of the automorphisms of the extension under consideration, and not any longer only as a group of permutations of the roots of an equation; and he proves (for the field of numbers) the fundamental theorem on the linear independence of automorphisms ([79], v. III, p. 29) as well as the existence of normal bases for a Galois extension ([79], v. II, p. 433). Finally, he tackles the problem of algebraic extensions of infinite degree, and notices that Galois theory can not be applied to it as it stands (an arbitrary subgroup of a Galois group not being always identical with the group of an extension with respect to a subextension); and by a bold intuition, he already begins to think of the Galois group as a topological group$^{30}$— an idea that will come to fruition only with the theory of Galois extensions of infinite degree, developed by Krull in 1928 [187 d].

Parallel to this evolution the notion of an element transcendental over a field becomes precise. The existence of transcendental numbers is proved for the first time by Liouville in 1844, by means of an explicit construction, based on the theory of diophantine approximations [204 c]; Cantor, in 1874, gives another "non constructive" proof using simple considerations about the power of sets [47]; finally, Hermite proves in 1873 the transcendence of $e$, and Lindemann in 1882 that of $\pi$ by a method which is analogous to that of Hermite, putting thus a final seal on the ancient problem of the squaring of the circle.$^{31}$

As for the role of transcendental numbers in algebraic calculations, Kronecker observes in 1882 that, if $x$ is transcendental over a field $K$, the field $K(x)$ is isomorphic to the field of rational fractions $K(X)$ ([186 a], v. II, p. 253). He makes besides the adjunction of indeterminates to a field the keystone of his exposition of the theory of algebraic numbers ([186 a], v. II, pp. 245-387). On the other hand, Dedekind and Weber show in the same year [80] how arithmetical methods can help to set up the theory of algebraic curves. One sees thus appearing in several directions analogies between Arithmetic

$^{29}$ The word "field" is from Dedekind himself; that of "ring" was introduced by Hilbert (Dedekind called rings "orders"). $^{30}$ "...The set of these permutations makes up in a certain sense a continuous multiplicity, a matter that we will not go into more deeply here" ([79], v. II, p. 288). $^{31}$ Simple proofs of these theorems will be found for example in [163 a], v. I, p. 1.

and Algebraic Geometry, which will turn out to be extremely fruitful for the one and the other.

In all this research, the fields that arise consist of “concrete” elements in the sense of classical mathematics — (complex) numbers or functions of complex variables.$^{32}$ But already Kronecker, in 1882, takes full account of the fact (obscurely sensed by Gauss and Galois) that “indeterminates” only play the role in his theory of the basis elements of an algebra, and not that of variables in the sense of Analysis ([186 a], v. II, pp. 339-340); and, in 1887, he develops this idea, linked to a vast programme that aims at nothing less than recasting all mathematics by rejecting all that cannot be reduced to algebraic operations on the integers. It is on this occasion that, taking up an idea of Cauchy ([56 a], (1), v. X, pp. 312 and 351) who had defined the field $\mathbf{C}$ of complex numbers as the field of remainders $\mathbf{R}[X]/(X^2 + 1)$, Kronecker shows how the theory of algebraic numbers is completely independent of the “fundamental theorem of algebra” and even of the theory of real numbers, all fields of algebraic numbers (of finite degree) being isomorphic to a field of remainders $\mathbf{Q}[X]/(f)$ ($f$ an irreducible polynomial over $\mathbf{Q}$)([186 a], v. III$_1$, pp. 211-240). As is remarked a few years later by H. Weber [327 a], developing a first sketch of the axiomatic theory of fields, this method of Kronecker is applicable in fact to all base fields $K$. Weber indicates in particular that one can take as $K$ a field $\mathbf{Z}/(p)$ ($p$ a prime number), thus making the calculus of congruences “modulo $p$” return to field theory; this latter had been born in the second half of the XVIIIth century, with Euler, Lagrange, Legendre and Gauss, and the analogy that it showed up with the theory of algebraic equations was not missed by observers; developing this analogy, Galois (with his research on group theory in mind) had not hesitated to introduce “ideal roots” of an irreducible congruence modulo $p$,$^{33}$ and had indicated its principle properties ([123], pp. 113-127).$^{34}$ When Kronecker’s method is applied to $\mathbf{Z}/(p)$, what is found again however (up to terminology) is the presentation

$^{32}$ No more than their predecessors, neither Kronecker nor Dedekind and Weber really define the notion of an “algebraic function” of one or more complex variables. One cannot indeed define correctly an “algebraic function” of a complex variable (in the sense of Analysis) until the corresponding Riemann surface is defined, and it is precisely the definition of the Riemann surface (by purely algebraic means) that is the goal pursued by Dedekind and Weber. This apparent vicious circle disappears of course when one defines a field of “algebraic functions” as an abstract algebraic extension of a field of rational functions: indeed it is only this definition that is used by Dedekind and Weber, which makes their results completely legitimate.

$^{33}$ In a manuscript dating in all likelihood from 1799, but published only after his death, Gauss, still without introducing “ideal roots”, obtains, in an equivalent form, a good part of the results of Galois ([124 a], v. II, pp. 212-240, in particular p. 217).

$^{34}$ Galois was fully aware of the formal character of algebraic calculations, not hesitating, for example, to take the derivative of the first member of a congruence to show that this latter does not have multiple “imaginary” roots ([123], p. 117). He underlines in particular that the theorem of the primitive element is valid as much for a finite field as for a field of numbers ([123], p. 117), without however giving a proof.

that had already been given by Serret and Dedekind ([79], v. I, p. 40) of the theory of these "Galois imaginaries".

To all these examples of "abstract fields" there come to be added, at the turn of the century, some fields of a new very different type, the fields of formal power series introduced by Veronese [318], and above all the $p$-adic fields of Hensel [157 f]. It is the discovery of these latter that led Steinitz (as he says explicitly) to bring out the abstract notions common to all these theories, in a fundamental work [294 a] which can be considered as having given birth to the actual concept of Algebra. Developing systematically the consequences of the axioms of commutative fields, he introduces thus the notion of prime field, of separable (algebraic) elements, of perfect fields, defines the degree of transcendence of an extension, and proves finally the existence of algebraically closed extensions of arbitrary fields.

In recent times, the theory of Steinitz has been completed in several important areas. On the one hand, the work of Artin has brought into prominence the linear character of Galois theory [7 a]. On the other hand, the general notion of derivation (copied from the formal properties of classical differential Calculus) sensed by Dedekind ([79], v. II, p. 412), introduced by Steinitz in the special case of a field of rational fractions ([294 a], pp. 209-212), has been used with success in the study (essential for modern Algebraic Geometry) of transcendental extensions, and notably in the generalisation to these latter of the notion of separability [330 d].