CHAPTER XIV.

INSOLUBILITY OF THE ALGEBRAIC QUINTIC.
Another of the famous problems in the history of mathe- matics, which long proved an ignis fatuus to mathematicians, is the solution of the general algebraic equation of the fifth degree. By a solution of an algebraic equation we mean the expression, by a finite number of radicals and rational functions, of a root of it in terms of its coefficients. The solution of an algebraic quadratic equation presents but little difficulty. In the sixteenth century, solutions of the general cubic and quartic equations were obtained. It did not seem unnatural to suppose that by similar analytic methods the solution of quintic equations, as also those of a higher order, might be effected. This is now known to be impossible, and I propose to give a brief sketch of the reason why it is so. The proof rests on the fact that the equations x^ = \ and oc^—l have no common complex root.
Quadratic, cubic and quartic equations can be solved by various methods ; but, in effect, all of them reduce the solution of the particular equation to the solution of one, called a resolvant, of a lower order. These methods fail when applied ■^o equations of an order higher than four. Lagrange was the earliest writer to ask whether this was necessarily the case. In 1770 and 1771 he published a critical examination of the known solutions of quadratic, cubic and quartic equations, and showed that such solutions were possible only because a function of the roots of such equations could be formed which had a smaller number of possible values than the order of the equation, and this function was such that its value could be determined.
328 INSOLUBILITY OF THE ALGEBRAIC QUINTIG [CH. XIV
For instance, take the quadratic equation x"^ -{■ acc-\-h = 0. It has two roots Xi and x.2. We can reduce the solution to that of a simple equation because we can form a function of ^1 and X2 which has only one value, and this value can be deter- mined. Such a value is y = (x^ - x^y, since (xi — x.^'^ = (x^ — Xif. Moreover we have y = (xi — x^Y = (x^ + os^Y — 4}X^X2 — a^— 46. Thus y is known. Hence the roots of the quadratic are given by
Xl "T X^ ^ Q/y
Xi X2 ^^ V2/* Similarly, in the cubic equation x^ + ax^ ^hx + c==0, if y = (^1 + ft>^2 + roots of unity, then y is a function of the roots which has only two values, namely (^1 + (0X2 + co^x^y, which may be also written in the form (X2 + cox^ + (o'^x^y or (x^ + cox^ + co^x^y, and (x2 + coxi + (o^Xsf, which may be also written in the form (^3 + (0X2 + oy^x^y or (xi + (oXs + (o^Xzy. Let y and z denote these two values, that is, y = (xi + (0X2 + (o^XsY, and z = (x2 •{• (oxi + co'^x.^y. Then y-^z=-2ci'-^9ab-27c=A, say, and y2={a^-Sby=B, say. Thus y and z are the roots of t^ — At + B — 0, an equation which can be solved. And the roots of the cubic are determined by
X^ "T" X2 ~T~ X^ -— ~" Ctf
Xl + (0X2 + w^a^s = \/y,
WXi +X2+ CO'X^ = i^/z.
Again, in the quartic equation x* + ax^ + bx'^-\-cx-\-d~0, if y = (xi — X2 + X3 — x^y, then 3/ is a function of the roots which has only three values, namely, (xi— X2+ x^ — x^y or (a?2 — ^1+^4 — ^3?, (xi — X3 + x^ — X2y or {xs — Xx-\- X2 — ^4)^ and {x^ — x^-\r X2 — x.^^ or (^4 — Xi-\- Xs — x^f. If these values be denoted by y, z, and w, we have y-^z-\-\i — Sa^ — 86 = ^, say, yz-\- zu-{-uy= 3a* - 16a-6 + 166" + 166c - 64(^ = -B, say, and yzu = (a^ - 4a6 + 86)- = G, say. Hence y, ^r, u are the roots of the equation P — At^ ■\- Bt -^ C = 0. And the roots of the quartic are determined by
Xy-^- X2-\- x^-\- x^ = — a,
a^i - ^2 + ^3 - ^4 = "^y*
Xi-X2 — x-i-\-x^ = V^,
Xi-]-X2 — X-^ — Xi= VUc
CH. XIV] INSOLUBILITY OF THE ALGEBRAIC QUINTIG 829
Lagrange showed that for an equation of the ??th degree, an analogous function y of the roots could be formed which had only n — 1 values, and which led to a resolvant of the degree n — 1, but that the coefficients of this latter equation could not be obtained without the previous solution of an equation of the degree {n — 2) !. Hence the form assumed for y did not provide a solution of a quintic or of an equation of a higher degree. But though he suspected that the general quintic and higher equations could not be solved algebraically, he foiled to prove it. The subject was next taken up by P. Ruffini, 1798 — 1806, but his analysis lacked precision.
The earliest rigorous demonstration that quintic and higher equations cannot be solved in general terms was given by N. H. Abel in 1824, and published in Crelles Journal in 1826. The result was interesting, not only in itself, but as definitely limiting a field of investigation which had attracted many workers. Abel's proof was simplified by E. Galois* in 1831, and is now accessible in various text-books. Essentially the argument is as follows.
Let rci, Xr,, x^, ... Xn be the roots of the equation
f{x) = x^-\-ax"-^+...+k = 0.
Any one of these roots, ajj, is a function (which will generally involve radicals) of the coefficients a, b, .... These coefficients are symmetrical functions of the roots, namely, a = Xx^, h = 'ZxiX2, .... If these values of a, h, ... be substituted in the expression for x^ we get, on simplification, an identity. If we interchange Xi and x.^, or any pair of roots, this will remain an identity. Similarly it will remain an identity, if we permute cyclically an odd number of roots, since such a permutation is equivalent to an interchange of a number of pairs of roots. For instance, if Xi and x.2 are the roots of the quadratic equation a^-{-ax-{-b = 0, we have
o^i=h[-a + V{a2 - 46}]
= i [(x^ + x^) + V{(^i + ^2)' - ^J^i^^a}]
= i [(^1 + ^2) + (^1 - ^2)].
* See Liouville's Journal, Paris, 1846, vol. xi, pp. 417—433.
330 INSOLUBILITY OF THE ALGEBRAIC QUINTIC [CH. XIV
This is an identity, and if we interchange x^ and x.^ it will remain an identity.
Now suppose that we have a solution of the equation, that is, an expression for oc^ in terms of the coefficients, involving only a finite number of radicals and rational functions. This expression may be a sum of a number of quantities. To fix our ideas let us suppose that in one of these quantities we come first, in the order of operations, to a radical, say, the 2?th root of H, where we may without loss of generality take p as prime. Of course H is rational: it involves a, 6, c, ..., and therefore is a symmetrical function of x^, X2, .... Thepth root of H also will be rational, but it will not be symmetrical : let us denote it by (f){xi, x^, ... Xn). For instance, in the quadratic equation H=a' — 4:h = {xi + x.,)"^ — 4^1^72, which is a rational and symmetrical function of x^ and x^. But i\/H = is not symmetrical, though it is rational.
In the general case (j> is rational but not S}Tnmetrical. It involves the two roots, Xi and X2, and therefore it must change in value if they are interchanged. Further, since the values of (j) are determined by ^^ = H, and H does not vary when the roots are interchanged, one of the values of (f> must be deducible from the other by multiplying it by co, where w is a pih root (other than unity) of unity, that is.
For instance, in the quadratic equation x- + ax-hh = 0, we observe that, if in a?i and a^g, we necessarily get ^{x.2, x^ — — two values of ^ = H, where H is invariable. Hence if one is + »JH, the other must be + ^/H The relation thus reached,
(p \X2, Xif X2, . ..^ = 0)Cp \Xif X2, X^y . . .Jf
is an identity; hence if we interchange Xi and Xr,, we have
(j>(Xi, X2, Xs, ...) = 0X^(572, a^i, Xs, ...).
Hence &)*=!. And as co^l, we have a) = — 1. Since &)- = !, we have p = 2: this shows that in the expression for a root of
CH. XIV] INSOLUBILITY OF THE ALGEBRAIC QUINTIC 331
an equation the first radical which occurs in the order of opera- tions must be of the second degree.
In the case of a quadratic equation this concludes the dis- cussion, for there are only two roots which can be interchanged. We may note that if it were possible to take the value &> = 1, it would at once give x._, = x^, which, in the general case, is clearly impossible.
We proceed to the case of a cubic or higher equation. We will first suppose that in the expression for x^ we substitute the above value of (/>, and combine it and similar square roots with the various rational functions of the coefficients, a, 6, .... As long as we only introduce such square roots we obtain a function of the roots, say K, susceptible of taking only two values, and therefore invariable when three (or any odd number) of the roots are permuted cyclically. This cannot lead to the determination of three or more roots. Hence we must, in this process of reduction and simplification, arrive, in the expression for a^i, at a radical, say the 5th root of /i, of a higher order than a square root. In this expression K will be invariable when three (or any odd number) of the roots are permuted cyclically.
We can express the 9th root of -ST as a rational function of the roots "^{xi.x^, ... a?„), and, from the nature of the case, "^ takes different values when three roots are permuted cyclically. The values of y^ are roots of -v/r? = K. Accordingly, following the same argument as that given above, we have
Y \X2, X^y Xij ... Xji) = CO 'Y' {^Xif X^f X-^, ... Xn)y
where o) is a 5th root (other than unity) of unity. If we permute x^, x.2, x^ cyclically, we have
T \'^3 J '^1 > ^) ^ii . . . ) = O) Y' yXo , X^, Xiy X^, • '•)}
and -^ (xi, x^, x^, Xt, ...) = a)-v/r(a'3, x^, x^, x^, ...).
Hence co^ = 1. We have previously shown that the first radical involved in the general expression for a root must be a square root, and now we see that the next radical must be a cube root. We observe that the solution usually given of a cubic confirms
332 INSOLUBILITY OF THE ALGEBRAIC QUINTIC [CH. XIV
our analysis. The solution of the cubic a^ -\-bx + c = 0 is generally written as
x={- c/2 + sju]^ + {- c/2 - y/u\^,
where u = cV^ 4- h^/27. In each term of the expression for x the first surd which occurs in the order of operations is a square root and the next surd a cube root.
In the case of a cubic equation there are only three roots, and we cannot continue the process further. So also we cannot proceed further in the case of a quartic equation, for as we want to permute an odd number of roots cyclically we cannot permute more than three.
We proceed to the case of an equation of the fifth or higher degree. We have already shown that in this case if, when we substitute in the expression for x^ the value of (/> and similar square roots, we arrive at a radical ^^K for which the equivalent rational function yjr takes different values when an odd number of the roots are permuted cyclically, it follows that if we per- mute three roots cyclically we get co^ =1, where o) is a root (other than unity) of w^ = 1. Hence q = S. Also if we permute five roots cyclically, we obtain by a similar argument o)" = 1. Thus q = 5. These equations for co and values of q are incon- sistent. In fact the argument shows that the first surd which has to be calculated in the general expression for Xi is a square root, and the next surd is at the same time a complex cube root of unity and a complex fifth root of unity. This is impossible. Hence an equation of the fifth or higher degree cannot be solved by a finite number of radicals and rational functions of the coefficients.
It may be added that just as we can express the root of a cubic equation in terms of trigonometrical functions, so we can express the root of a quintic or sextic equation in terms of elliptic or hyperelliptic functions. But such functions lie outside the field of algebra.
333