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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
