508 EQUATION efficients. Reverting to the before-mentioned particular equation x* + x 3 + x 2 + x + 1 = 0, it is very interesting to compare the process of solution with that for the solu tion of the general quartic the roots whereof are a, I, c, d. Take GO, a root of the equation o> 4 - 1 = (whence CD is 1, - ], i, or- i, at pleasure), and consider the expression (a + ub + tu 2 c + ul) 4 , the developed value of this is a 4 + & 4 + c 4 + d* + 6(aV + bW) + I2(a-bd + Pea + <?db + d -ac) c 3 + d*a) + 1 2(a-cd + Pda + c*ab + d-bc) } + c"d 2 + cZ 2 a 2 ) + 4(a :! c + b*d + c 3 + d*b) + 24abcd} + w 3 { 4(a cJ + Va + c*b + d*c) + 12(a 2 k + b"cd + cla + d*ub)} that is, this is a 6-valued function of a, b, c, d, the root of a sextic (which is, in fact, solvable by radicals; but this is not here material). If, however, a, b, c, d denote the roots r, r 2 , r 4 , r 3 of the special equation, then the expression becomes r 4 + r 3 + r + r 2 + 6(1 + 1 ) + 12(?- 2 + r 4 + r 3 + r) + w {4(1+1 +1 +1 ) + 12(r l + ?- 3 + r +r-)} + or[6(r + r ! + r 4 + r i ) + 4 (r- + r 4 + r" + r )} + w 3 {(r + r 2 + r 4 + r 3 ) + 12(?- 3 + r + r- + I A ] } viz., this is = - 1 + 4eu + 14cw 2 - 16&) 3 , a completely determined value. That is, we have (? + w>- 2 + orr 4 + o, 3 r 3 ) 4 = - 1 + 4co + 14<o 2 - 16c* 3 , which result contains the solution of the equation. If w=l, we have (r + r 2 + r* + r 3 ) 4 - 1, which is right; if w = - 1, then (r + r 4 - r 2 - r s ) 4 = 25; if co = i, then we have {r-r^ + i (?- 2 -r 3 )} 4 = -15 + 20t; and if w=-i, then {r-t A -i (/2-r 3 )) 4 ^ -15-20J; the solution may be completed without difficulty. The result is perfectly general, thus : n being a prime number, r a root of the equation x n ~ l + #"~ 2 ....+#+1 = 0, to a root of oj"~ 1=0, and g a prime root of a n ~ l L 1 (mod. n), then (r + uro . . . . +w-V"~ 2 ) ~ 1 is a given function M + M,co ... +M n _ 2 a) n ~ 2 with integer coefficients, and by the extraction of (n - l)th roots of this and similar expressions we ultimately obtain r in terms of w, which is taken to be known ; the equation x n - 1 = 0, n a prime number, is thus solvable by radicals. In particular, if n - 1 be a power of 2, the solution (by either process) requires the extraction of square roots only ; and it was thus that Gauss discovered that it was possible to con struct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting developments in regard to the theory were obtained by Jacobi (1837) ; see the memoir " Ueber die Kreistheilung, u.s.w.," Crelle, t. xxx. (1846). The equation x n ~ l + . . + x -t- 1 = has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by Abel (1828), and since called Abelian equations, viz., for the Abelian equation of the order n, if x be any root, the roots are x, Ox, 2 #, . . . Q n ~ l x (Ox being a rational function of x, and O n x x} the theory is, in fact, very analogous to that of the above particular case. A more general theorem obtained by Abel is as follows : If the roots of an equa tion of any order are connected together in such wise that all the roots can be expressed rationally in terms of any one of them, say x ; if, moreover, 6x, 6^x being any two of the roots, we have 6@^x Ofa, the equation will be solvable algebraically. It is proper to refer also to Abel s definition of an irreducible equation : an equation (J>x = Q, the coefficients of which are rational functions of a certain number of known quantities a, b, c , . , is called irreducible when it is impossible ts express its roots by an equation of an inferior degree, the coefficients of which are also rational functions of a, b, c . . . (or, what is the same thing, when $x does not break up into factors which are rational functions of a, b, c . . ). Abel applied his theory to the equations which present themselves in the division of the elliptic functions, but not to the modular equations. 32. But the theory of the algebraical solution of equations in its most complete form was established by Galois (born October 1811, killed in a duel May 1832 ; see his collected works, Liouville, t. xi,, 1846). The definition of an irre ducible equation resembles Abel s, an equation is re ducible when it admits of a rational divisor, irreducible in the contrary case ; only the word rational is used in this extended sense that, in connexion with the coefficients of the given equation, or with the irrational quantities (if any) whereof these are composed, he considers any number of other irrational quantities called "adjoint radicals," and he terms rational any rational function of the coefficients (or the irrationals whereof they are composed) and of these adjoint radicals ; the epithet irreducible is thus taken -either absolutely or in a relative sense, according to the system of adjoint radicals which are taken into account. For instance, the equation # 4 + # 3 + x 2 + #+1=0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x 2 + x + 1 ) 2 |# 2 , and it has thus the irrational divisors # 2 + -i(l + x /5)#+ 1, x~ + l(l -J5)x+l; and these, if we adjoin the radical A/5, are rational, and the equation is no longer irreducible. In the case of a given equation, assumed to be irreducible, the problem to solve the equation is, in fact, that of find ing radicals by the adjunction of which the equation be comes reducible ; for instance, the general quaclric equation x z + px + q = Q is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical -v/i/ >2 ~ <! The fundamental theorem is the Proposition I. of the " Memoire sur les conditions de resolubilite des Equations par radicaux ;" viz., given an equation of which a, b, c . . are the m roots, there is always a group of permutations of the letters a, b, c . . possessed of the following properties : 1. Every function of the roots invariable by the substi tutions of the group is rationally known. 2. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group. Here by an invariable function is meant not only a function of which the form is invariable by the substitu tions of the group, but further, one of which the value is invariable by thess substitutions ; for instance, if the equa tion be cj>x = 0, then 4>x is a function of the roots invariable by any substitution whatever. And in saying that a func tion is rationally known, it is meant that its value is ex pressible rationally in terms of the coefficients and of the adjoint quantities. For instance, m the case of a general equation, the group is simply the system of the 1.2.3 ... n permutations of all the roots, since, in this case, the only rationally determinable functions are the symmetric functions of the roots. In the case of the equation x n ~ l ...+#+1 = 0, n a prime number, a, b, c . . . k = r, r 3 , r a<1 . . . r 9 "" 2 , where g is a prime root of n, then the group is the cyclical group abc . . . , be . . . ka, . . . kab . . . j, that is, in this particular case the number of the permutations of the group is equal to the order of the equation. This notion of the group of the original equation, or of the group of the equation as varied by the adjunction of a series of radicals, seems to be the fundamental one in Galois s theory. But the problem of solution by radicals,
instead of being the sole object of the theory, appears as the