EQUATION 507 the solution of algebraical equations, be found that they all ultimately depended upon finding a " resolvent" equa tion of which the root is a 4- o>6 4- ore + u> 3 d 4- . . . , u> being an imaginary root of unity, of the same order as the equation; e.g., for the cubic the root is a + o>?> + o> 2 c, a> an imaginary cube root of unity. Evidently the method gives for L 3 a quadric equation, which is the "resolvent " equa tion in this particular case. For a quartic the formulae present themselves in a some what different form, by reason that 4 is not a prims number. Attempting to apply it to a quintic, we seek for the equation of which the root is (a + wb + arc 1 + w 3 tZ 4- w 4 e), a> au imaginary fifth root of unity, or rather the fifth power thereof (a 4- coi 4- arc + oAZ 4- o>V) 5 ; this is a 24-valued function, but if we consider the four values corresponding to the roots of unity w, w-, to 3 , w 4 , viz., the values (a + <a b + o> 2 f + <Sd + a> 4 c) 5 , (a + <u-b + a> 4 c + co d + &> : e) 5 , (a 4 co :i 6 + w C 4- <a 4 d + ca^e) 5 , (a ~ u> 4 6 + &/ ! c -t- urd + ia c-Y , any symmetrical function of these, for instance their sum, is a six-valued function of the roots, and may therefore be determined by means of a sextic equation, the coefficients whereof are rational functions of the coefficients of the original quintic equation; the conclusion being that the solution of an equation of the fifth order is made to de pend upon that an equation of the sixth order. This is, of course, useless for the solution of the quintic equation, which, as already mentioned, does not admit of solution by radicals ; but the equation of the sixth order, Lagrange s resolvent sextic, is very important, and is intimately con nected with all the later investigations in the theory. 29. Tt is to be remarked, in regard to the question ct solvability by radicals, that not only the coefficients are taken to be arbitrary, but it is assumed that they are represented each by a single letter, or say rather that they are not so expressed in terms of other arbitrary quantities as to make a solution possible. If the coefficients are not all arbitrary, for instance, if some of them are zero, a sextic equation might be of the form x c> 4- b.i A + ex- 4- d = 0, and so be solvable as a cubic ; or if the coefficients of the sextic are given functions of the six arbitrary quantities a, l>, c, J, e, f, such that the sextic is really of the form (x 2 + ax+b)(jc 4 + cx 3 + dx n - + ex+f) = Q, then it breaks up into the equations x 2 4- ax 4- b - 0, z 4 4- ex 3 4- dx* 4- ex 4-/= 0, and is consequently solvable by radicals; so also if the form is (,e - a) (x - I) (s - c) (x - d) (x - e) (x -f) = 0, then the equation is solvable by radicals, in this extreme case rationally. Such cases of solvability are self-evident ; but they are enough to show that the general theorem of the non-solvability by radicals of an equation of the fifth or any higher order does not in any wise exclude for such orders the existence of particular equations solvable by radicals, and there are, in fact, extensive classes of equa tions which are thus solvable; the binomial equations x" - 1 = present an instance. 30. It has already been shown how the several roots of the equation x n - 1 = can be expressed in the form 2sT SST cos J -+i sin . , but the question is now that of the ?t ?( algebraical solution (or solution by radicals) of this equa tion. There is always a root = 1 ; if o> be any other root, then obviously w, or, . . . . a/ - 1 are all of them roots ; x n - contains the factor x-1, and it thus appears that W, w 2 , . . . a/ 1 " 1 are the n - 1 roots of the equation we have, of course, a/ " 1 4- w"~ " . . . 4- w 4- 1 = 0. It is proper to distinguish the cases n prime and n com posite ; and in the latter case there is a distinction accord ing as the prime factors of n are simple or multiple. By way of illustration, suppose successively n= 15 and n~ 9 ; iu the former case, if a be an imaginary root of x 3 - 1 = (or root of x 2 + x+ 1 =0), and ft an imaginary root of z 6 - 1 = (or root of x 4 4- x 3 4- x 2 + x + 1 = 0), then w may be taken = aft ; the successive powers thereof, a/3, a- ft 2 , ft 3 , a/3 4 , a 2 , ft, a/? 2 , a 2 3 , ft a, a 2 # /2 2 , a/3 3 , a 2 /? 4 , are the roots of x u 4- ./, 13 .... 4- #4- 1 = ; the solution thus depends on the solution of the equations x 3 - 1 = and ar 5 - 1 = 0. In the latter case, if a be an imaginary root of x 3 1 = (or root of jo- 4- x 4- 1 = 0), then the equation # 9 - 1 = gives a: 3 1, a, or a 2 ; x 3 = 1 gives x = 1, a, or a 2 ; and the solu tion thus depends on the solution of the equations x 3 - 1 = 0, a^-a = 0, 3 -a 2 = 0. The first equation has the roots 1, a, a 2 ; if ft be a root of either of the others, say if ,8 3 = a, then assuming o> = /3, the successive powers are ft, ft 2 , a, aft, a/5 2 , a 2 , a?ft, a 2 /8 2 , which are the roots of the equation x s 4- x 1 . . . 4- x + .1 = 0. Ic thus appears that the only case which need be con sidered ia that of n a prime number, and writing (as is more usual) r in place of w, we have ?, r 2 , r 5 , . . . 7 """ 1 as the (n - 1) roots of the reduced equation then not only r n - 1 = 0, but also r"" 1 4- r"~ a ... 4- r 4- 1 = 0. 31. The process of solution due to Gauss (1801) depends essentially on the arrangement of the roots in a certain order, viz., not as above, with the indices of r in arith metical progression, but with their indices in geometrical progression; the prime number n has a certain number of prime roots g, which are such that g n ~ l is the lowest power of <7, which is - 1 to the modulus n ; or, what is the same thing, that the series of powers 1, </, g z . . . g n ~ 2 , each divided by n, leave (in a different order) the remainders 1, 2, 3 . . . n - 1 ; hence giving to r in succession the indices 1 , y, ff n ~ 2 , we have, in a different order, the whole series of roots r. r 2 , r 3 . . . r"" 1 . In the most simple case, n = 5, the equation to be solved is x 4 + x 3 + x- + x 4- 1 = ; here 2 is a prime root of 5, and the order of the roots is r, r 2 , r 4 , r 3 . The Gaussian pro- I cess consists in forming an equation for determining the periods P 15 P , = r + r* and r 2 4-? 3 respectively, these being such that the symmetrical functions P 1 4-P , PjP are rationally determinate : in fact P x 4- P 2 = 1 , P^ = ( r + r 4 )(r 2 + r 3 ), = r 3 + r 4 + r 4- r 7 , = r 3 + ? 4 + r + r", -1. P 1? P 2 are thus the roots of w 2 4-M-l=0; and taking them to be known, they are themselves broken up into subperiods, in the present case single terms, r and ? i4 for P I} r 2 and r 3 for P., ; the symmetrical functions of these are then rationally determined in terms of P a and P ; thus ? > 4-r 1 = P 1 , ?r 4 =l, or r, r* are the roots cf M 2 - P t ?f 4- 1 = 0. The mode of division is more clearly seen for a larger value of n; thus, for = 7 a prime root is = 3, and the arrangement of the roots is r, r 3 , r 2 , ? %6 , ? 4 , r 5 . We may form either 3 periods each of 2 terms, P I} P 2 , P., r + ) <6 > r 3 4. r 4^ r 2 4 ? .5 respectively ; or else 2 periods each of 3 terms, P ; , P 2 = r + r 2 + r 4 , r 3 + r 6 + r> respectively ; in each case the symmetrical functions of the periods are rationally determinable : thus in the case of the two periods P : 4- P 2 = - 1 , P^ = 3 4- r 4- r 2 + r 3 + r 1 4- r 6 + / e , = 2 ; and the" periods being known the symmetrical functions of the several terms of each period are rationally determined in terms of the periods, thus r + r 2 + ?- 4 = P 1 , r. ?- 2 4- r. r* + r 2 . t A P.,, r.r-. r 4 = 1 . The theory was further developed by Lagrange (1808), who, applying his general process to the equation in ques tion, x n ~ l 4- z n ~* . . 4- x 4- 1 = (the roots a, b, c . . being the several powers of r, the indices in geometrical progression as above), showed that the function (a 4- w6 4- arc 4- . .) n ~ i
was in this case a civen function of to with, integer co-