Algebraic Solution of an Let the equation man .. au -=.0 (I) have a group G for a certain domain R, whose factors of composition r,, . . .
are all primes. Then (1) can be solved alge braically. For the corresponding chain of re solvents =0, 4'3=0, ...
have groups of prime orders r,, rt, . . ; they are therefore cyclic equations, whose solution has just been effected. Since, as will be set forth later at more length, it is never necessary to employ other than rational resolvents, the above results leads to Galois' Criterion for the Solution of an Equation by Radicals. In order that (1) admit an algebraic solution it is neces sary and sufficient that the factors of composi tion of its Galoisian group consist of primes only.
Application to the Solution of the Biquadro tic x4+ ch.rs + ch.r* (us + a. — O. (11) For simplicity let us suppose its coefficients are independent variables. Let the original domain of rationality R embrace besides the coefficients a cube root of unity p. Then the group of (11) is the symmetric group S.. As subgroups of S. we note the alternate group which con sists of all the substitutions of S. which can be
obtained by an even number of exchanges of the roots of our equation, the axial group 11, (12)(34), (13) (24), and the semi-axial group G2--- 1I, (12) (34) } . The groups S., A., G., G2, I forrn a series of compo sition whose factors are obviously 2, 3, 2, 2.
As they are primes, the equation (11) admits an algebraic solution. To solve (11) let us proceed with Starkweather as follows: To form our first resolvent, let us use the subgroup /12, and take as function belonging to this group 9$=--(xi - x2) (rz -x2) (xi- x4) (xi- x2) (x. - x2) (x,-- x2) (12) The corresponding resolvent is (13) where is the discriminant of (11).
On adjoining (p----v.iour domain is Ri(Rd 4), for which the group of (11) is 244.
A subgroup of A 4 iS G4 A rational function. belonging to this is 11,==xix2 This gives the resolvent "I'=110-0244+ a4(alt - 4at) arl --= O. (14) The solution of this cubic, which is a cyclic equation, gives 4, as a known explicit function of quantities in R.. On adjoining IP we get the domain Rs' R. V :474) , for which the group of (11) is G.. The next subgroup we take is G2 to which belongs