GALOIS' SOLUTION OF AN EQUATION.
Let G be the group of the equation &an + Limn + . . . + an = 0 (I) for the domain R. Let H. be a subgroup of G of index Let fqxi, . . .xn) be any one of the infinity of rational functions belonging to H.. Then (fri is root of a rational resolvent .1(v)=0 of degree ri. On solving 4%-=0 and adjoining one or more of its roots to form a new domain R., the group of (1) is now a sub group of GI of G. Let H. be a subgroup of Gi of index ro, to which belongs the rational func tion 02(xi, . . . xn). This is the root of a re solvent$,(y) =0 of degree r,. On solving and adjoining one or more of its roots to form a new domain R,, the group of (1) is now a subgroup G. of G1. As the order of the groups G, G,, G,, decreases, we must eventually arrive at the identical group when the roots of (1) are rationally known. Since the group G usually admits quite a variety of subgroups, and since the functions belonging to a given group are infinite in number, Galois' theory shows that the number of ways for solving a given equation is endless. At the same time it clearly shows that the number of distinct ways is usually quite limited, depending on the sub groups of G.
Among the solutions of the equation (1) which Galois' theory offers, one class is par ticularly interesting, depending on a Series of is defined as follows: Let G. be an invariant subgroup of G, such that G contains no invariant subgroup containing G,. It is then a maximum invariant subgroup of G. If G has no maximum in variant subgroup besides the identical group, it is simple. The series of groups G, . (10) such that each is a maximum invariant sub group of the preceding group, is called a series of composition of G. If the index of Gm under Gm—, is rm, the numbers r,, ... c, are called the factors of composition. It may he possible to decompose a group G into a series of com position in more than one way. Thus the cyclic group C., 1, s, si, where s= eafx.x.xl.x.) =(l, 2, 3, 4, 5. 6) ad xononorrer, • mits the series C., A,1
and C., B, 1, where A=-- 1 1, The factors of composition of the first series are 2, 3, while those of the second series are 3, 2. They are thus the same aside from their order. A theorem of Jordan states that how ever a group be decomposed in a series of com position, the factors of composition are the same aside from their order.
What makes the solution of an equation by means of a series of composition so remarkable is the fact that the resolvents .1— 0. . . , OA = 0 corresponding to the subgroups G1, G, ...GA of have groups T., ... FA for their respective domains which are simple. Their orders are the factors of composition. Moreover, any root of one of these equations is a rational function of any root of that equa tion. Thus on adjoining one of its roots the same effect is produced as adjoining all. Finally, the resolvent equations 4=0 are the simplest possible.
Cyclic Equation of Prime When the group G of an equation F(x) is a cyclic group of prime order p its solution is readily effected, as Abel show ed. Let the roots of F=0 be xo, Xp—,, and let 1, p— 1). Then G= 11.).Y=,.. .
For the case in hand we may suppose the pth roots of unity p, p', . . . lie in the orgi nal domain of rationality. Consider the rational functions fia=z0+ ...+ h=-1, On applying y they go over into Hence are unaltered by 7 and hence by G They are therefore rationally known. On ex tracting a pth root we get x.-i- phx1+ . .
This system of p —1 equations together with x0+ x,+. . .
gives 1 P—I P FA Xa — p , g =0, 1, . . . h=0 The pth roots which enter here must be mined uniquely in terms of one of them, say The others are rational in this one, for (x.-1-phx1+ . . .+ P:c1+ remains unchanged for y and hence for G. Hence these AA are rationally known. We have now Xe= (fsli gia AA.
Pes h p" This result gives the theorem: Cyclic equations of prime degrees can be soloed algebraically, i.e, by the extraction of roots from known quanti ties.