THE SOLUTION OF THE QUINTIC.
We have seen that the equation of fifth de gree Q=0 whose group is the symmetric group cannot be solved by means of radicals, i.e., by resolvents of the type X in On adjoin ing (xi—xi) (x,—x.) (xi—x.) (xr—xi) (24---V4) (x1-x.) (x.-x.) the square root of its discriminant .1, the group of Q=0 reduces to the alternate group As of 60 substitutions. But As is simple. Thus is a simple equation for the domain R(V4). Other algebraic equations having this group arise in the theory of linear differential equa• Lions, and also in the theory of elliptic func tions. In fact the hypergeometric function a-p a.a 1.p-p +I F(a, 13, y, x+ x,+ 1.y 1.277+1 is a solution of a very simple differential equa tion of the second order G. For variable a p, y, it represents a new transcendent; but for certain values of these parameters it re duces to the elementary functions; e.g., it may become algebraic. In seeking for these latter cases Schwarz was led to introduce a new vari able s, the quotient of two fundamental inte grals of G 0. This variable for certain values of a, 13, y satisfies the equation I (s)1728x f H' where I.
The equation stands in so intimate relation with the icosahedron that it is called the icosa hedral equation. Indeed if we project stereo graphically the icosahedron, on the s-plane, the centre being at the origin, the 12 vertices and the middle points of the 20 faces will be pre cisely the roots of f and H respectively.
From this it is easy to conclude that the group of 1:1 is formed of the 60 rotations which leave the icosahedron unchanged. Klein has shown that the icosahedral equation whose roots are very simple known functions of F(a, i9, y, x can be put in connection with The equation .1=;) may thus be considered as a normal resolvent of the quintic.
A normal resolvent which springs from the elliptic functions is the following: In trigo nometry one of the problems is to express sin — in terms of sin x, n a prime number. This
may be done algebraically, as is readily shown. In the elliptic functions the same problem arises. Here the algebraic relation between P to,,o,) and p (u, ir,, (00 is of degree n'—l.
The solution of this equation depends upon an equation of degree n+1 called an equation of transformation. For n=5 such an equation is 4'yl-FlOztyl-12g.31+5, (15) whose group is the above As and whose roots are and (0 2o, -1-48ro, 464+596r1 -.a 1, 2, 3, 4.
Here 41 is the discriminant g,'---27a. How equations of this type could be set in relation with the quintic was first shown by Hermite in 1858. The equation (15) was used by Kie pert. It forms a very convenient normal re solvent of the quintic.
Having found in the elliptic functions con venient normal resolvents for this quintic, we might hope to employ the equations of transfor mation of higher orders to solve the general equations of higher degrees. The considera tion of their groups, however, shows very easily that this is not possible. To find suitable equations we must pass from the elliptic to the hyperelliptic functions. By their aid the gen eral equation of every degree can be solved.
Bolza, 0., (Theory of Sub stitution Groups and its Application to Algebra' (American Journal Math. V. 13); Cajori. F., (Theory of Equations' (New York 1904) ; Hil bert, article in Crelle's Journal (V. cx) ; Wil der, article in Mathematische Annalen (V. xxxiv) ; Jordan (ibid., V. I, 1869) (Traite des substitutions' (1870) ; Netto, (Substitutionen theorie' (tr. by Cole, F. N.) ; Pierpont, Theory of Algebraic Equations) (Annals of Mathematics, V. I, II, 1899-1900) ; Serret, (1866) ; Weber, (Algebra' (2d ed., 1898-99).