FUNCTIONS,

SERIES

AND METHODS

IN ANAL YSIS

Burnside, W. S. and Panton, A. W. 1881, The Theory Df Equalions, 1st edn, Dublin: Hodges, Figgis, and London: Longmans, Green. [AIso later edns. Influential textbook; contains historical notes.] Fourier, J. B. J. 1831, Analyse des équations indéterminées (ed. C. L. M. H. Navier), Paris: Firmin Didot. [German transls: 1846, Braunschweig: Meyer; 1902, Leipzig: Engelsmann (Ostwalds Klassiker, No. 127).] Hamburg, R. R. 1976, "The theory of equations in the 18th century: The work of Joseph Lagrange', Archive for History of Exact Sciences, 16, 17-36. Lagrange, J. L. 1798, Traité de ia résolution des équations numériques [... ], Paris: Duprat. [2nd edn 1808; also in his Oeuvres, Vol. 8.] Runge, C. 1898, 'Separation und Approximation der Wurzeln', in Encyklopiidie der mathematischen Wissenschaften, Vol. I, 404-448 (article I B 3a). Sinaceur, H. 1991, Corps et modeles [... ], Paris: Vrin.

4.11

Solving higher-degree equations U. BOTT AZZINI

1 INTRODUCTION The proof first given by Paolo Ruffini 1799 and, independent!y of him, by Niels AbeJ 1826 of the impossibility of algebraic solution of the general quintic equation settled a question which had been opened by the work of the Italian algebraists of the Renaissance (§6.1). For some three hundred years mathematicians had thought that the solution of the fifth-degree equation could be given 'by radicais', as was the case for the equations of degree 2 to 4. ln such cases the solutions were expressed by formulas involving only rational operations on the coefficients and extractions of roots. Their guiding idea was that a similar method must exist (and therefore had to be found) for the general algebraic equation of degree n. ln other words, it had to be always possible to reduce an nth-degree equation to the simple form x" = A by means of auxiliary equations of degree (at the most) n - 1 (Pierpont 1895). Joseph Louis Lagrange 1770-71 proved that for n > 4 the auxiliary equation (the 'resolvent') was of a degree strictly greater than n (§6.1)~ Using Lagrange's result, Ruffini showed that the quintic could not be reduced to the form XS = A by solving algebraic equations of lower degree. It was possible, however, to reduce the equation to the form f(x,A) = O, f being a fifth-degree polynomial with coefficients given by rational functions of a parameter A. Therefore, x could be algebraically expressed as x = g(A), and the general quintic equation could be solved by means of radicais involving the function g. The reduction of the fifth-degree equation to the form f(x, A) = O had actually been found independentiy by the Swedish mathematician E. S. Bring in 1786. Some fifty years later, in 1834, the Englishman G. B. Jerrard showed that it was always possible to put an algebraic equation of degree n into such a form that it did not contain terms of degree n - 1, n - 2 or 566

567

til

FUNCTIONS.

SERIES

AND METHODS

IN ANALYSIS

SOLVING

n - 3. To this purpose he used the foIlowing method of transformation, introduced by Ehrenfried Tschirnhaus in a paper of 1683: given the

HIGHER-DEGREE

EQUATIONS

§4.11

actually found a way of expressing the roots of the quintic equation means of elliptic functions. He considered the reduced form

equation

x5

x- a= O

-

by

(8)

(1)

and its auxiliary equation xn-l = b1xn-2 eliminate

x between equations

+ ... + bn-2X + bw-: + y,

(2)

(1) and (2) to obtain

of

a new equation

and asked whether it was possible to represent each solution of ít by means of single-valued functions of new variables. The analogy of the trigonometric solution of the cubic equation suggested to him that he should consider transcendental functions analogous to them (i.e. elliptic functions). Hermite considered the moduli k and k:' of the elliptic integrais (§4.5)

degree n, (3) where the coefficients c» depend on the b«. Tschirnhaus had thought (wrongly) it was always possible to determine the bk in such a way that Ck = O for k = 1,2, ... , n - 1, thus reducing (after the transformation) equation (3) to the form yn + c; = O. For n = 5 Jerrard showed that, by using Tschirnhaus's method (i.e. solving equations of degree 2 and 3 only), the general equation could be reduced to the form x5+x+a=0. This is the Bring-Jerrard researches

form,

(4)

which

was the starting-point

of the

K=

t 1 /2

1

o

OF THE QUINTlC

Before Hermite published his result in 1858, an important step had been m~de by Enrico Betti in 1854 using the foIlowing reduction form:

+ 5x3

x5 (y being a parameter).

=y

By differentiating 5x2(x2

After eliminating x between equation calculations, he obtained the equation dy 5(y2 + 108)112

(5) he found that

= dy.

(6) 2

(5) and x

x2dx 4 6 (x + 4x - 8x2

+3=

O and some more

+ 12)112'

The left-hand side of equation (7) is easily integrable by functions, while the right-hand side can be transformed into differential by means of the change of variable x2 = z. Therefore, the equation could be solved (at least in principie). Independently of him, the same idea was pursued by Hermite 568

dé

(9)

(1_k'2sin2<1»1I2'

(7) elementary an elliptic Betti said,

of power

!J;(w)=~k'.

(10)

+ v6 + 5U2V2(U2

(w) = [
-

v2)

+ 4uv(1

- U4V4)

= O.

(11)

the expression

+
x [
][
+ ~'16)

-
+;

'16)

(5)

equation

+ 3) dx

o

If n is a prime number, then v =
Hermite then considered

EQUATION

K' =


q=exp(i1tw),

u6

SOLUTlON

and

As Jacobi had shown, ~k and ~k' can be expressed as quotient series of q = exp( - 1tK'1 K). Hermite then considered

of Charles Hermite.

2 HERMITE'S

I

1t12

d (1- k2sin2<1»1/2

l

(12)

Now, (w+ m '16), m = O, 1, ... ,4, are actually the roots of the following quintic equation whose coefficients can be rationally expressed by means of 5 - 24.

53<1>
= C.

(13)

Hermite was able to show that equation (13), under the assumptions he made for (w), = {/(24. 53)
(14)

1858, who 569

!

ii

=::

FUNCTIONS,

SERIES

AND METHODS

IN ANALYSIS

SOLVING

and therefore 2 a

(15)

By assuming cp4(W) = k as an unknown, and recalling that y,4(W) = k:' (with k'? = 1 - k2), he obtained the quartic in k k4 +A2k3

+ 2k2

-A2k+

1

= O,

(16)

A =~(55)a/2,

which alIowed him to determine k and consequently (8).

EQUATIONS

§4.11

~een w~rking ~n the same problem. Two months after Hermite had published his solution, Kronecker communicated his own method in a letter to him published in the Comptes Rendus of the Paris Academy of Sciences. Like Hermite, Kronecker had first folIowed the idea of using the modular equation and the reduction of the quintic to the form (8). He then decided inste~d to adopt a more direct procedure. he considered a cyclic (rational) function f(xo, ... ,X4) of the five roots xo, ... ,X4 of the quintic as

1 + cp8(w)

= {f(55)cp2(w)y,4(w)'

HIGHER-DEGREE

the roots of the quintic

sin(2~1t)

mtonto

(XmX~+nX~+2n

+ jJX~Xm+nXm+2n),

(20)

and thought that could be determined in such a way that the 12 transformed functions obtained by means of even permutations of the x's could be written as x f, ±fo, ±fl, ±fz, ±f3, and ±f4 in order to satisfy both the relation

'I

,I I

I

!

"

ii

:! i

Ii '~

,p .~

jJ

3 THE

WORK

OF BRIOSCHI

AND

KRONECKER

lnspired by Hermite's paper, in the same year of 1858 Francesco Brioschi published a second method for the solution of the quintic equation. Instead of using the modular equation, as Hermite did, he started by considering the 'equation of the multiplicator', which had been introduced by Carl Jacobi in his work on the transformation of elliptic integrais. In 1827-8 Jacobi had solved the problem of determining a rational function y such that dy (P4(y»1/2

dx = (P4 (x»

f2

dx - k2x2)]

1/2

Aldy = [(1 _ y2)(1 _ À 2y2)]

1/2'

r" - IOcpf6

(18)

(19)

=O

(21)

+ 5'1'2 = y,f2,

(22)

where 'I' and y, are rational functions of both the coefficients and the square root of the discriminant of the quintic. Then he observed that equation (22) could be reduced to J acobi's 'equation of the multiplicator', and therefore solved by means of elliptic functions.

where the unknowns are the coefficients of U and V, the new module À, and AI (calIed the 'multiplicator' of the transformation of order n). For n = 5, z = l/AI satisfied Jacobi's 'equation of the multiplicator' (z-I)6-4(z-I)5+28k2k'2z=0.

+ fã + fr + f~ + fj + f~

and the equation

whcre P4(X) is a fourth-degree polynornial (§4.5). He was able to show that, for every odd prime n and any module k, the problem reduced to the determination of a rational function y = U(x)/ V(x) (U being a polynomial of degree n, and V apolynomial of degree n - 1) such that [(1 - x2)(1

fi

I'

r

,~

Ili

(17)

112 '

í: I'

-,

Kronecker was not entirely satisfied with his result because of the necessity of introducing what he called an 'accessorial irrationality', the extraction of the square root of the discriminant of the quintic in order to ensure the condition (21). As Abel had done for the equations of lower degree (2 to 4), Kronecker wanted to use only 'natural irrationalities', auxiliary quantities which depended on the roots of the quintic in a rational way. In other words, he was looking for a method in which the auxiliary quantitíes he needed remained in the field of the roots of the equation. But later he discovered that it was impossible to obtain an equation like (22) without introducing 'accessorial' quantities, nor to obtain a resolvent of the equation depending on one parameter only. In other words, it was ímpossible to reduce the general quintic equation to a form depending on one parameter only without leaving the field of the roots. Kronecker's claim was to be proved by Felíx Klein in 1879.

Just as Hermite had done, Brioschi was able to obtain from equation (19) a fifth-degree resolvent which could be reduced to the Jerrard form (8) after some more calculations based on the properties of Jacobi's equation (19). FolIowing a suggestion by Hermite, Brioschi quickly realized that equation (19) has a simpler resolvent whose existence, according to a method elaborated by Leopold Kronecker, was to be ensured a priori. For some years, independently of Hermite and Brioschi, Kronecker had

The method, which Kronecker communicated in a cryptic way in a letter to Hermite, was explained in full detail by Brioschi in 1858 and by Kronecker himself in 1861. A complete discussion of the different methods (including alI the calculations they entailed) was eventually given by Hermite in 1866.

570

571

FUNCTIONS,

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IN ANA L YSIS

4 LA TER DEVELOPMENTS Kronecker had elaborated his method of tackling a more general problem concerning seventh-degree equations. This problem was eventualIy solved in 1878by KIein, who showed that it was possible to reduce alI seventh-degree equations, whose Galois group was a certain simple subgroup of order 168 of the alternate group A7, to the modular equation of order 7, and then to solve them by means of elliptic functions. In the late 1870s Klein's researches on higher-degree equations intertwined with Brioschi's before they found their natural setting in the theory of automorphic functions (§3.16). Klein himself focused on the deep connections between the algebraic aspects of the theory of equations and the geometrical ones in his book 1884 on the icosahedron, and gave an historical account of the previous developments and presented in geometrical terms the main concepts of the theory of the quintic equation (Slodowy 1986).

BIBLIOGRAPHY Abel, N. H. 1826, 'Beweis der Unmõglichkeit algebraische Gleichungen von hõheren Graden aIs dem vierten alIgemein zu lõsen", Journal für die reine und angewandte Mathematik, 1,65-85. [French transl. in Oeuvres completes, 2nd edn, 1881, Vol. 1, 66-87.] Hermite, C. 1858, 'Sur Ia résolution de l'équation du cinquieme degré', Comptes Rendus de l'Académie des Sciences, 46, 508-12. [Also in Oeuvres, Vol. 1, 5-12.] Klein, F. 1884, Vorlesungen über das Ikosaeder und der Gleichungen vom fünften . Grade, Leípzig: Teubner. [Repr. 1992, Basel: Birkhãuser.] Lagrange, J.-L. 1770-71, 'Rétlexions sur Ia résolution algébrique des équations', Nouveaux Mémoires de l'Académie Royale des Sciences de Ber/in, (1770), 134-215; (1771), 138-253. [Also in Oeuvres, Vol. 3, 203-421.] Pierpont, J. 1895, 'Zur Geschichte der Gleichung des V. Grades (bis 1858)', Monatshejte für Mathematik und Physik, 6, 15-68. Ruffllii, P. 1799, Teoria generale delle equazioni in cui si dimostra impossibile Ia soluzione algebraica delle equazioni genera/i di grado superiore aI quarto, Bologna: S. Tommaso d'Aquino. [Also in Opere matematiche, Vol. 1, 1-324.] Slodowy, P. 1986, 'Das Ikosaeder und die G1eichungen fünften Grades', in H. Knõrrer et aI., Arithmetik und Geometrie, Basel: Birkhãuser, 71-113.

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