THE FUNDAMENTAL THEOREM OF ÁLGEBRA BEFORE CARL FRIEDRICH GAUSS

This is a paper about the first attempts of the demonstration of the fundamental theorem of algebra . Before, we analyze the tie between complex numbers and the number of roots of an equation of -n-th degree . In second paragraph we see the relation between the integration and fundamental theorern . Finally, we observe the linear differential equation with constant coefficients and the Euler's position about the fundamental theorern and then we consider the d'Alembert's, IJuler's and Laplace's dernonstrations . lt is a synthesis paper dedicated to Pere Menal a collegue and a friend .

Niccoló Tartaglia2 .1n his book it appears for the first time an special quadratic equation : But, as Remmert remembers us, "it is not clear whether Cardano was led to complex numbers through cubic or quadratic equations" 5 .The sense of these words is the following : while quadratic equations equations If some one says you, divide 10 into two parts, one of which multiplied into the other shall produce 30 or 40, it, is evident this case or equation is impossible3.
The narre imaginary is introduced by René Descartes, as we will see soon .But it is debt, perhaps, to following Cardano's words : " . . .you will nevertheless imagine -15 to be the difference between . ..", completing, in that case, the square .
See Bombelli, 13. .[9, 1691 or Bertolotti reprint, 133 .9 Bombelli did not through too much on the nature of cornplex numbers, but he knows, for example, that 3 2f11i=2fi .We can now match coefficients in the two forms T2 = 4 and 3 = 4 cos 30, so that A _ With this value of A; we can select a value of 0 so that cos 30 = 4 q = q/2 (p/3)3 In the casos in-eductibilis, we have 0 = (2)2 -( 3 ) 3 < 0 and then <1 and thus the condition for three real roots ensures us that 1 cos 301 < 1, which is essentiall3 .
In 1637 René Descartes wrote La Géométrie".This appendix was his only mathematical work; but a what work!1t contains the birth of analytic geometry 15 .In Book III of his La Géométrie Descartes gives a brief summary of that was known about equations 16 .Between his 13 Then he proves the equivalence : we have cos 30 = u, where M = " 72 Civen te, (p/3)3 .
algebraic assertions l7 , we are interested in the following : in every equation there are as many distint roots as is the number of dimensions of the unknown quantities 18 .
This is an important approach to Fundamental Theorem of Algebra, but it is not the first and perhaps never the more explicit .
The first writer to assert that "every such equation of the nth degree has n roots and no more" seems to have been Peter Roth`.The law was next set forth by a more prominent algebraist, Albert Girard, in 1629: Every algebrasc equation admits as many solutions as the denomination of the highest quantity indicates . . .20 Girard gives no proof or any indication of one.He merely explains his proposition by sorne examples, including that of the equation x4 -4x + 3 = 0 whose solutions are 1, 1; -1 + if ~ .

21
;-1-if r7'1'he otlier irnportant assertions in Book 111 of La Géométrie are : -A polynornial P(x) whicli vanisiies at c is always divisible by the factor x-e and tren [This theorem was probably already known by Thornas Later another mathematician, named Rahn [or Rohnius], also gave a clear statement of the law in his Teutschen Algebra [66] .
The question about these formulations of the Theorem is the following : these algebraists accepted real and complex numbers and only them as solutions of equations?The answer is not easy nor clear.Girard accepts the "impossible solutions" with these words Someone could also ask what these impossible solutions are .I would answer that they are good for three things : for the certaintly of the general rule, for being sure that there are no other solutions, and for its utility 22 .
Descartes, by his sido, realized the fact that an equation of the nth degree has exactly n roots23 .But, for Descartes ; the imaginary roots do never correspond any real quantity24 .[19] ; who not only stated the law but distinguished between real and imaginary roots and between positivo and negativo real roots in making the total number", for Remmert [Remmert, R .[68,100]], contrarily, "Descartes takes a rather vague position on the thesis put forward by Girard" .22 Girard, A .[38] in Viéte and alü [83,141] .In other sido [Viéte and alii [83,142]] he says : "Thus we can give three names to the other solutions, seeing that there are some which are greater than nothing, other less than nothing, and other enveloped, as those which have V/'--, like V"--3 or other similar numbers ." Remmert, R. [68,99], goes further .He says: "He thus leaves open the possibility of solutions which are not complex" .Remmert thinks that, in his ambiguity, Girard leaves an open door to the solutions more cornplicated than the complex .The problem consists to know the exact sense of the Girard's words "iTrapossible solutions" because, for him, "there are no other solutions" .[About this question see also Gilain ; C .[37,[93][94][95] .]23 This assert is debt to the Descartes' text [see Descartes, R .[19,380] .English translation in Smith, D .E.-Latham, M .[75,175]] : Neither the true nor false roots are real ; sometimes they are imaginary; that is, while we can always conceive of as niany roots for each equation as I have already assigned, yet three is not always a defanite quantity corresponding to each root so conceived of.Thus, while we may conceive of the equation x3 -6x 2 +13x-10=0 as having three roots, yet three is only one real root, 2, while the other two, however we may increase, diminish, or multiply them in accordance with the roles just latid down, remains always imaginary .
In this text there is a rather interesting classification signifying that we may have positivo and negativo roots that are imaginary.
It seems that for Descartes the roots are always real or imaginary and no other kind of root is possible .[about with this oppinion, see Gilain, C .[37, 95- 168525 .He says: "The Geometrica,l Effection, therefore answering to this Equation may be this" 2s .
[76, 1. ] .In a letter to Collins, May 6, 1.673, Wallis suggests a construction a little different from arny of the constructions found iri his Algebra [see Cajori, P .[13]] .We shall see this alternative construction liere:

. The technique of integration and complex quantities
The eighteenth century use of the integral concept was limitad .Newton representad the transcendental functions as series arld integrated these functions term by term21 .Gottfried Willielm Leibniz and Johann Bernoulli treated the integral as the inversa of the differential 30 .
In this context the decomposition of rational fractions [or functions] into partial [or simple] fractions made possible a decisivo stop in integral calculus31 .
The problern was calculate the integral where P and Q are polynomials and deg(P) < dog Q and, for getting it, Gottfried Wilhelm Leibniz and Johann Bernoulli, together other trlatherrlaticians of his time, saw the necessity to express every real polynomial as product of real factors of first and second degree32 .Tltis fact shows us that tliey liad very much confidente in the Fundarnental Theorem of Algebra33 .
representation .28 The satisfactory geometrically representation of complex quantities was carried by the Norwegian mathernatician Caspar Wessel in 1797 and independently by the Swiss Jean Robert Argand in 1806 .T his lasa work, despite its considerable merit, rernained unnoticed until a Frencli translation apperead in 1897 .29 See Pla, .I .[64,[9][10][11][12][13][14][15][16][17][18][19][20] .30 See Kline, M .[44,406] : "If dy = f'(x) dx, then y = f(-c) .'lhat is, a Newtonian antiderivative was chosen as the integral, but differentials were used in place of Newton's derivatives" .31 The existente of an integral was never questioned .32The Arilhrnetica Universalis of Isaac Newton contains, as we have said before, the substance of Newton's lecturas from 1673 to 1683 at Cambridge .In it are fourid many important results in equations theory, such as the fact that the imaginary roots of a real polynornial "nrust occur in conjugate pairs" .This fact is a very important result and it was naturally accepted by the mathematicians of the and of seventeenth century .But, following Leibniz, this fact presents difficulties, as we shall sea next .33 See Leibniz, C .W . [49] ; [51] and Bernoulli, Ih .[6] .
The: chance did that in 1702, July 10, Johann Bernotilli, thinking to enunciate hirri a new result, wrote to Leibniz that liad found tlie integral of differential quantities q dx, where p and q are polynomials .But Leibniz responded : "No only 1 llave already the solution of this problern, but rnoreover 1 llave it from the first years in which 1 practicad the higher geometry .In this result 1 llave seen an essential comporieril; of be transformed also [using now z = b 2 ] into the differential of "a sector or circular arc -2 t dt ta and reciprocally" .Finally he observes that the integral of b2 +z 2 depends on the quadrature of the circle, and moreover which are two differentials of imaginany logarithms : one sees that imaginary logarithms can be taken for real circular sectors because the compensation which imaginary quantities malees on being added together of destroying themselves in such a way that their sums is always real 37 .
We have observed there the introduction of imaginary logarithmic differential into the integration of rational functions38.
38 We do not explain the history of imaginary logarithms .But theie are many papers on complex logarithins as, for example, Cajori, F .But this situation is not easier than it seerns .In his presentation about the integral of rational functions, Leibniz shows us a difficulty, a limitation or merely a question.It is always possible decompose a real polynomial into a product of real lineal factors or real quadratic factors? 39or, every polynomial has always a real and complex root and, with every complex root, has also the conjugate complex root?Although always Leibniz is clear and rotund when he says As soon as I had found my Arithmetic Quadrature; reducing the quadrature of circle into a rational quadrature and observing that the sum depends of the quadrature of the circle, I immediately observed that a time reduced to the summation of a rational expression, all quadrature can be converted in many kinds of summation of the more simple .And I will show, by a decomposition proceeding of a new genes because it must be in this manner.This proceeding consists to convert a product of factors into a sum; this is, to transform a fraction with a denominator of higher degree, egaall to product of roots, into a sum of fractions with simple denominators, 4°w hen he rnust integrate f~h e finds a problern .It is possible obtain a,,+ a4 to multiply ~+ a2 by -x7---% -a2 , but they are not real.And it is not possible to obtain a real decomposition ; because 39 This assert is absolutely clear in Newton, 1 .[59], -as we have seen in the footnote 32 .40 Leibniz, G .W . [50,[351][352] .
In this work Leibniz obtains naturally the integration of rational functions, as for example although " J .~is the quadrature of the hyperbola'' .
Next year Leibniz studies the case in which the roots are not simple and therefore the sum is transformated into the sum of fractions with multiple denominators [see Leibniz, G .W . [51]] .
arad therefore it is not possible to reduce f x4 +a4 to the quadrature of the circle nor to the quadrature of the hyperbola .It would be necessary to introduce the quadrature of J , ~as a new function 42 .
There is neither hesitation about the importante which Leibniz granted t11e complex numbers and his contributions, "when they were almost forgotten", were remarkable41 .Between these it is interesting to observe that he obtained an imaginary decomposition of a positive real number which surprised his contemporaries and enriched the theory of lmaginaries : Then b 2 + = 1+vr--3 + 1-v/---344 .41 Leibniz does not observe that x4 +a 4 = [.22 + avf2x + a21 .¡x2 -a. ,,í2x + a2] .
The possible mistake is debt to have begun by the complex conjugate decomposition 42 We have already introduced the quadrature of the hyperbola f -x * a and the quadra- Cure of circle .lx ~.Then, says Leibniz, "1 wait that we will be able to follow this --z dx a-d.m progression and we will found the problems related with f x , f g, . . ." [see Leibniz, G .W . [50,3601] .43 Moreover, for Leibniz, complex numbers are Che natural consequence of have accepted real numbers : "From the irrationals are born the impossible or imaginary quantities whose nature is very strange but whose usefulness is not to be despised" [see Leibniz, G .W . [50,51] Moreover ; as says Boyer, "Leibniz did not write the square roots of complex numbers in standard complex form, nor was he able to prove his conjecture that if f (z) is a real polynomial."4s  Finally in an unpublished Leibniz's paper46 appears the so-called de Moivre's formula.He does not explain how he found it, but it is comprehensible to us as where x = COSO, y = cos é47 .But these important mathematical contributions did not enough to clarify the nature and reality of the complex numbers .Leibniz adventures his mistic nature, saying : "The nature, mother of the eternal diversities, or the divino spirit, are zaelous of her variety by accepting one and only one pattern for all things .By these reasons sha has inventad this elegant and admirable proceeding.This wonder of Analysis, prodigy of the universe of ideas, a kind of hermaphrodite between existente and non-existente ; which we have named imaginary roots" as .
This mysterious character stood during several centurias, may be until the Euler's time with the contributions of the own Euler and d'Alembert .
Kline is absolutely clear in this sense: Complex numbers were more of a bine to the eighteenth-centrl,ry mathematicians .These numbers were practically ignorad from their intvoduction by Cardan until aboutt 1700 .Then complex numbers were usad to inteyrate by the methode of partial fractions, which was followed by the lengthy controversy about complex nnmbers and the logarithms of negativo and complex nnmbers.Despite his correct resolution of the probleni of the logaráthms of complex nurnbers, neither Euler nor the other, ntathematicians were clear about those numbers.
Euler tried to understand rahat complex numbers really are, and in his " Vollstündige Anleitung zur Algebra", which frst appeared in Russian ira 176'8-6.9 and in Germany in 1770 and, is the best algebra text of the eighteenth century, says, Because all conceivable numbers are either greater than zero or less than 0 or equal to 0, then it is clear that the square roots of negativo numbers cannot be included among the possible numbers [real nurnbers] .Consequently we must say that these are irnpossible nnmbers .And this circumstance leads us to the concept of such nurnbers, which by their nature are irnpossible, and ordinarily are callad imaginary or fancied numbers, Because they exist only in tire imagination Euler made 7nistakes with, complex nurnbers .In this Algebra he writes v'-1 = v/4 -= 2, Because ~/a, -\íb -= /_ a b .He also gives i, 2 = 0 .2078795763,but misses other valv,es of this quantity`~a .

The three first attempts to prove the Fundamental Theorem of Algebra
One possible enunciate of the Fundamental Theorem of Algebra 5°is : Every polynomial P(x) vith real coefficients has a cornplex root.But I think that, historically, this distintion is not clear .The former mathematicians to Gauss was not conscious of that fact .51 Gauss considered the Theorem so important that he gave four proofs ; the principles on which the first is based was discovered by Gauss in October 1797, but the proof was not published until 1799 .In this proof, similar to d'Alembert's attempt of proof, he does not introduce cornplex numbers.He proves the Theorem in the forro : Every polynomial P(x) urith real coefficients can be factored into linear or quadratic factors .
The second and third proofs of Theorem were published in 181 .6 .The second proof is purely algebraic, following perhaps the Euler's intention.The forth proof is based in the same principle of the first and was published in 1849 .In this proof Gauss uses already cornplex numbers more freely because, he says, "they are now common knowledge" .In the third proof he used, in fact, that what we today know as the Cauchy integral theorem .
A half century dedicated by Gauss to prove the Theorem .
Following these different demonstrations we can find precisely the differences noted by Gilain . 52 The Euler and Lagrange attempts were published, respectively, in 1751 and 1774 .53 Pierre Simon Laplace made an attempt to prove the Theorem, quite different from the Euler-Lagrange attempt but also algebraic, in his Legons de mathématiques donnés a l'Ecole Norrnal, published in 1812 .
Really therefore was Euler the first of three three mathematicians which userted the true of the Theorem .So ül a letter to Nikolaus Bernoulli,' Euler ennuniates the factorization theorem for real polynomials, closing the question poned by Leibniz 54 .54 We have already seen that "does not seein to have ocurred to Leibniz that f could be of the forro a + b i, because if he had seen that he would have noticed that the product of the factors [X +a-,,/¡,] -[X -{-av'-] and [X -a,/¡,] -[X -av--] are both reals and then he would have obtairied So he wuold have avoid hin mistake .l t is remarkable that he should not have ¡)con led to this factorization by the simple advice for writing X 4 +0.
See also Kline, 1VI .[44,[597][598] : " . . .Leibniz did not believe that every polynomial with real coefficients could be decomposed into linear and quadratic factors .Euler took the correct position .In a letter to Nikolaus Bernoulli of October 1, 1742, Euler afñrmed without proof that a polynomial of arbitrary degree with real coefficients could be so expressed [see Euler, L .[1.8621, 1, 5251 .Nikolaus did not believe the assertion to be correct and gave the example of -4x 3 +2x 2 +4x+4 with the itnaginary roots 1 after assert that he doubted once when he saw this example, did it doubt once seen the example, "pointed out the complex roots occia-in conjvgate pairs, so the produt of x -[a + b i ] and x -[ab i ], wherein a + bi and ab i are a con,jugate pair, gives a quadratic expression with real coefficients .Euler then showed that hin was true for Berrioulli's example .But Goldbach, too, re.jected the idea that every polynomial with real coefficients can be factored into real factors and gave the example x 4 -h 72x -20 [see the letter from Goldbach to Euler of february 5, 1743 in Euler, L .[27,1,193]] .
Euler then showed Goldbach that the later liad rnade a rnistake and that he [Euler] liad proved this theorem for polyriomials up to the sixth degree .I-Iowever, Coldbach was not convinced, because Euler did not succeded in giving a general proof of this assertion" .
The reader interested to follow the succession of these letters can see, for example, Gilain, C .[  (3] A+Br+Cr2 +---+Lrn =0 .In fact, the general solution depends of the factorization of the polynomial [3] and of the nature of its roots -reals or complex ; simple or multiple-, and indirectly his result depends essentially of the Fundamental Theorem56 .In order lo reduce in general a differential rational function to the quadrature of the hyperbola or lo that of the cirele, it is necessary, according to the rnethod of M. Bemoulli [Mero.Atad . Paris, 1702], to show that every rational polynomial, without a divisor composed of a variable x and of constants, can always be divided, when it is of even degree, tinto trinomial factors xx + fx + g, xx -l-hx -1-i, etc ., of which all coeficiente f, g, h, i .
. .are real .It is clear that this diiculty affects only the polynornial that cannot be divided by any binomial s: -{a, x -}-b, etc ., because we can always by divison reduce to zero all the real binomials, if tivo are any, and it can easily be seen that the products of there binoinials will give real factors xx -1f x -f-g [see Struik, D .J .(  Then, if 1 and 2 are true and xo is the point in which ~P(x)j atteints the minimum, then IP(xj = 0.This is the sketch of the d'Alembert's proof 60 . The second step is, for d'Alembert, the more important" and the proof offered by d'Alembert depends essentially on the Newton's method 59 D'Alembert accepts without demonstration the step 1 and the Newton's method .A simple elementary proof of d'Alembert lem?na was given by Argand in 1806 .This mathematician was one of the co-discoverers of the geometric representation of complex numbers .He represents the complex numbers as a vectors ¡rito the plan .Then J .PLA I CARRERA P(x) =a~x'+a,,-¡x"-i+ . ..+alx+ao is a vector OA,+1 .The demonstration consists to see that it is possible to choose x such that the point A,,+1 coincides with O .By seeing this, he explains where A is constant and ¡El is srnall compared to I0 xI when 10 xI is srnall .
Then, choosing the adequate direction of vector 0 x, it is possible obtain that A 0 x was opposite in direction to P(x0) .Theri
D'Alembert examines the set of real inrages y and takes the mininmrn yo which associate x is of complex form.Brrt following Che development (*), all real number y very Glose to yo must be also in image of the cornplex nurrrbers x.Tllen, if yo ,-E 0, there is an irnage closer to zero than yo .Contradiction .This contradiction establishes the Tlieorem.
It is interesting to note two innportant facts which were observed by d'Alembert ¡rito his work.Tlre first are corollaries 1 and 11 and proposition 111 6 `1 and says : "if a complex number a + b -,/_ -1 is a root of the polynornial P(x), then a-b x/_ -1 is anotlrer rooot of P(x) and tllen P(x) can always be decomposed into quadratic factors of the kind xx+'rnx+n" .

. The Euler-Lagrange's attempt .
The idea of Euler's dernonstration 67 was to decompose every mon¡c polynornial with real coefficients P(x) of degree 2" > 4 into a product 62 See Newton, 1 .[59] and Stillwell, J .[76,[125][126] .The sense of this theorem is the following: "To every pair (xo, yo) of cornplex numbers with yo -P(xo) = 0, there correspond an increasing series {qk} of rational numbers such that ek -[yyo]'k ¡ti a neighborhood ofyo'" .].'his theorem was proved rigorously by Pusierix in 1850 .lt is possible to avoid this theorem like we can see, for example, in D&rrie, H . [  But it is the forth theorem which gives us the key of his ideas : Every equation of the forth degree, as x4 +Ax 3 +Bx2 +Cx+D=0 can alvays be decomposed into two real factors in the second degree.
First, setting x = y -áA, he obtains that every equation of the forth degree can be of the form x4 + M x2 + N x + P = 0.If we decompose this equation in two equations of the second degree, we have 68This fact is also emploved by Laplace in his demonstration as we will see next .
70Then the polynornial has a factor of the form xx -I-px -f q .
Euler gives an example of how to decorripose an equation of the forth degree into two quadratic factors .
So Euler gives answer to the former problem posed by Nikolaus Bernoulli and Goldbach [see footnote 54] .7r We have there a partial proof of the Bolzailo-Cauchy theorem on InteT-mediate Value.
Thus, if P(x) is a polynomial of the form p(X) = x2"1 + B x2m-2 + C x 2m-3 + . . ."from which the value of u must be found.And since the absoluto terco -N -N is essentially negativo, we have hope that this equation has at least two real values 72 .Among the corollaries to Theorem 4 there is the statement that the resolution into real factors is now also proved for the fifth degree, and Scholiuln 11 points out that, if the roots of the given fourth-degree equation are xl, x2, z3, x4, then the sixth-degree equation in u, u being the sum of two roots of the given equation, will have the six roots X1+X2, xl+x3, T1+x4, z2+x3, x2+X4, X3+x4 .Since xl +X2+x3+X4 = 0, we can write for u the values ul , u2, u3, -ul, -712, -u3, and the equation in u becomes 12 When we take one of them as u, then the values of a and f3 will also be real, seeing that 73We can observe that the fourth roots XI, X2, i3, X4 of the equation Next to, tinto the theorem 5, he establishes Every equation of degree 8 can always be resolved tinto two real factors of the forth degree 74 .
The problem consists to see that not only u; but also the other cofficients a, /d; y, S, e, o are reals, a reasoning which Lagrange and, more later, Gauss objected.
Lagrange takes this equation but he observes that when u takes the value 0 into the rational expressions of the other coefficients of P (x) and P2(x) as fonction of u, it is possible obtain undefined coefficinets of the form 2 .For avoid this, he takes as unknown [when an = 1], v = 2n + a,,_1 and then observes that the "imaginary roots" of the 13is constant term is -ea1 u2 u3 .The product ui u2 u3 is real?There is .Euler does not explain this with detail .He says only that this product is real because the fundamental theorem of the theory of symmetric functions .
We can reasoning this : Despite this product was not a symmetric fonction of the symbols XI, x2, x3, x4 ; it is unvariable when we do all possible permutations of the roots of the equation [1], under the condition [2], between the roots of the equation [1] .Really this product can be obtained of the following : Remember that the fundamental theorem of the theoy/ of symmetrie functions says : Every rational fonction of roots of an algebraic equation IP(x1,x2, . .,xn) which.takes k different values when it makes all possible permutations of roots, satisfies an algebraic equation of degree k whose coeicients are rational functions of the coefcients of the given equation.
Then, if k = 1, the fonction W(x) satisfses a rational expression of the coefficients of the given equation.
Euler uses largely this fundamental theorem, but he only develop, with a sufficient rigour, for the general case of the second degree equations, but the theorem in 11is general form was proved firstly by Lagrange in his transcendental paper Reflexions sur la resolution algebrique des equations [1771] .So it will be necessary hope the Lagrange's apports by obtaining the general result .74 First the term x 7 is eliminated, so that the two supposed factors can be written x 4 -u x3 + cr x 2 +Q x +7 and x4 + u. x, 3 + 6 x 2 + e x + ?p .where a runs oven the set S", of all permutations of set {1, 2, . . ., n} .It is easy see that the produce of vo is always <_ 0. Next he avoids the case in which the product is zero, substituing vo for a useful combination of the coefcients of P1 with real coeflicients and then using his results contained in a paper of 1770-1771 7 on permutations of an equation, finishes rightly the (lemonstration 7e .
3 .3 .The Laplace's attempt . Za(k+,.) In the year 1795 ; Pierre Sirnon Laplace lnade an attempt to prove the Fundarnental Theoren177 .This attempt was completely algebraic, but quite different frorrt the Euler-Lagrange attempt .This mathematician and politician assumes, as his predecessors, that the roots of polynomials "exist" 78 .

Laplace says79
Of this it results a demonstration very simple of this general theorem which we have énnounced before and which, says that every equation of even degree can be solved into real factors of second degree .
The equation Qt (x) which roots are xi + xj + t (xi xj), where t E R arbitrary and i < j, has a degree of the form 2k-1 q', where q' E 2N+180.Then Laplace proceeds by induction on k: if k = 1, the new polynomial Qt (x) will have an odd degree and then it will be a least a real root xi + xj + t (xi xj) 81 .
It is clear that there is infinitely many real values t such that, for a same xi and xj, This quantity, "as we have seen" 82, can be solved in two real factors of second degree83 .
BOIts degree is exactely 2" q [2 k q -1]/2 = 2k -1 q', where « E 2N -{-1 . 81Laplace applies the following corollary of the Intermediate value Theomm: 'Tvery polynomial of odd degree has at least one real root" . 82See Laplace, P.-S .[47,[60][61][62][63] . 83Laplace considera the case in which the two factors Then the problem is finished because P(x) has a real factor of second degree iff every real equation of degree 2k-1 q', q' E 2N1+1 has a simmilar factor, and then [for the same reasorl] iff every equation of 2k-2 q", q" E 2N1 + 1 has a simmilar factor and following we establish the proof84.This fact is an easy consequence of the main theorem on symmetric functions which was proved by Newton in 1 .673 .This theorem says that the coefñcients of Qt(x) are real because "they are real polynomials in the elernentary symmetric functions of xl, 22, . . . .X," : that is, in the real numbers bt , . ., b,, .
Next year, in a very important paper55 , Euler thinks about the homogeneous nth-order differential equation with constant coefficients and finally, B; C, D, . . ., L are constants .He points out that the general solution of [1] must contain n arbitrary constants and the solution will be a sum of n particular solutions yj, every one multiplied by an arbitrary constant.So the general solution of y has the form y=CIy1+C2y2+ . .+C ;~yn-Then he makes in [1] the substitution y = e [f r dx] , with r constant, and obtains the polynomial equation in r,

55
Euler, L .[22] .56Each root ri of the polinomial equation [3] furnishes a partial solution into the sum [2] in accordance with the nature of each root rj , j = 1, . ..,n : -if rj is a real simple root of [3], then it furnishes into the sum [2] the sumand zj = Dj erj X ; -if rj is a multiple real root of multiplicity k, the k equal roots rj furnish into the sum [2] the sumand z j,k = eT' x ¡Do + Dl x + . . .+ D k-1 x k-1 -if rj = aj + ¡Oj is a simple complex root of [3], then it and its conjugate T .7 = aj -i/jj furnish into the sum [2] the sumand z i = e°j z [Di cos /3j x + DZ sin /3j x] -si rj = aj +iOj is a multiple complex root of multiplicity k, then the k equal roots rj = cej + i ej and their k conjugate roots furnish into the sum [2] the But, as we have already said, the first attempt of demonstration of the Fundamental Theorem of Algebra is debt to d'Alembert57 .3.1 .The d'Alembert's attempt .Really d'Alembert proves the existente of the root of P(x) in two steps58: 1 .There is the minimum xo of the module jP(x)j ; 2 .The d'Alembert's lemma: if P(x0) r,~0, then any neighborhood of sumand P=0 ea ¡ x x P [Di C tosOj x + DZ e sin PI x]Somewhat later [Euler, L .[24]] he treated the nonhomogeneous nth-order linear differential equation X(x) =Ay+Bdy-1-Cd22a .57 ll'Alembert remembers the Johann Bernoulli's text and then he says : "Nobody, what 1 know, have went more far [in the question of the decomposition of polynomials], if we exclude mister Euler, which in the tome VII of Miscellanea Berolinensia declares that he has demostrated the proposition in the general case .But 1 seem me that Euler never has published yet en this theorern [d'Alembert, J .le Rond [2, 183]] .5SSee d'Alerribert, J .le Rond [2] and Petrova, S .S .[62] .In the d'Alernbert's words : xo contains a point xr such that Ip(xl)j < jp(x0)j59 . .
. + aix + ao E C[x], there is a x0 E C such that IP(xo)j = inf I P P I . of polygon 62 .Applying this, d'Alembert obtains k>o Ck .[yyo]1k .

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This fact is essential in the first Gauss' demonstration [see, for example, Hollingsdale, S .[41,[319][320][321][322]] .o7 See Euler, L .[23] and Lagrange, J .-L .[45].P1(x) -P2(x) of two monic polynomials with real coefficients of degree m = 2n-1 .Then Euler asserts that p,. . . .are real functions in B, C, . . ., u, and that, by elimination of a, 9, 1 . .I A, N,, . . ., is obtained a monic real polynomial in tt of degree ( 2m ) whose constant term is m negative .Now this polynomial in u has a zero u by the intermediate value theorem as Euler clearly knew68.Now we can follow quickly the Euler's steps69 :1 .If the equation has a root of the form x + y xl-1, then there is also another of the form xy VI_ 70 ; 2. Every equation of odd degree has al least one root ; 3. Every equation of even degree with negative absolute term has at least one positive and one negative root71 .
have a common factor .This factor must be necessary a factor of the difference of two polynomials and then it must be Im(xi + xj) x + Im(xi xj) .If we divide P(x) by Chis polynornial of first degree, we will have a polynomial with odd degree and then it will have a real root r .The product[lm(xi + xj) x + Im(xi xj)] -[x -r]constitutes the factor of second degree found .8 'This proof has a mistake, like we can see in Remmert, R.[68, 122]  .It is necessary to see that the polynornial Qt(x) = H [x -(xi+xj)+t(xixj)]ER[x] 1<i<j<n [that is : all coeflicients are reals] .
Trisection of ;his angle gives us the solution z = cos B of the equation .Conversely, the problerrn of trisecting al] angle with cosine fa is equivalen ; to solve the cubic equation 4z3 -3z = p .1alt is, as it is well known, the third appendix of his famous Discours de la rnéthode pour bien conduire sa raison el chercher la verité dagas les sciences.The other appendices are La Dioptrique and Les Météors .For a comment we can seeBos, [32]J .M .[10],Milhaud,C.[56,,Pla,J .[63],orScott,J.F .[71,].1*STheanalytic geometry was independently discovered by Pierre Fermat ; a French amateur matliernatician, ¡ti his "Ad locos planos et solidos isagoge"[32] 97] .]The use of word imaginary in his actual sense begin here [see Smith, D .E .-Latham,M .[75, 175, footnote 207]] .24 Descartes confess that one is quite unable to visualizo imaginary quantities [see This impossibility or difficulty for visualizing irrlaginary quantities was perhaps the reason which carried the English mathematician John NVallis to give a geometrical interpretation in his Treatise of Algebra of Before 1799, year in what Karl li%iedrich Gauss gave his first rigorous proof of Fundamental Theorem of Algebra 51 , three important mathematicians had already made three attempts to prove the Theorem .The first is debt to a French mathematician and philosopher, Jean le Rond d'Alembert, and was published in 1748; but elaborated in 1746.Three years later, in 1749, Leonhard Euler gave an algebraic demonstration, very different of the d'Alembert's demonstration .This demonstration was completed by Joseph Louis Lagrange in 177252 .Several years later another French mathematician ; Pierre Simon Laplace, tried to prove the Theorem .lt was the year 179553 .