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, r4, r3 of the special equation, then the expression becomes
| r 4 | + r 3 + r + r 2 + 6 (1 + 1) | + 12 (r 2 + r 4 + r 3 + r ) |
| + ω{4 (1 + 1 + 1 + 1) | + 12 (r 4 + r 3 + r + r 2) } | |
| + ω2{6 (r + r 2 + r 4 + r 3) | + 4 (r 2 + r 4 + r 3 + r ) } | |
| + ω3{4 (r + r 2 + r 4 + r 3) | + 12 (r 3 + r + r 2 + r 4) } |
viz. this is
a completely determined value. That is, we have
which result contains the solution of the equation. If ω = 1, we have (r + r 2 + r 4 + r 3)4 = 1, which is right; if ω = −1, then (r + r 4 − r 2 − r3)4 = 25; if ω = i, then we have {r − r4 + i(r 2 − r3) }4 = −15 + 20i; and if ω = −i, then {r − r 4 − i (r 2 − r 3) }4 = −15 − 20i; the solution may be completed without difficulty.
The result is perfectly general, thus:—n being a prime number, r a root of the equation xn−1 + xn−2 + ... + x + 1 = 0, ω a root of ωn−1 − 1 = 0, and g a prime root of gn−1 ≡ 1 (mod. n), then
is a given function M0 + M1ω ... + Mn−2ωn−2 with integer coefficients, and by the extraction of (n − 1)th roots of this and similar expressions we ultimately obtain r in terms of ω, which is taken to be known; the equation xn − 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 construct geometrically the regular polygons of 17 sides and 257 sides respectively. Some interesting developments in regard to the theory were obtained by C. G. J. Jacobi (1837); see the memoir “Ueber die Kreistheilung, u.s.w.,” Crelle, t. xxx. (1846).
The equation xn−1 + ... + x + 1 = 0 has been considered for its own sake, but it also serves as a specimen of a class of equations solvable by radicals, considered by N. H. 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, θx, θ2x, ... θn−1x (θx being a rational function of x, and θnx = 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 equation 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, θx, θ1x being any two of the roots, we have θθ1x = θ1θx, the equation will be solvable algebraically. It is proper to refer also to Abel’s definition of an irreducible equation:—an equation φx = 0, the coefficients of which are rational functions of a certain number of known quantities a, b, c ..., is called irreducible when it is impossible to 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 φx 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.
24. But the theory of the algebraical solution of equations in its most complete form was established by Evariste Galois (born October 1811, killed in a duel May 1832; see his collected works, Liouville, t. xl., 1846). The definition of an irreducible equation resembles Abel’s,—an equation is reducible 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 x4 + x3 + x2 + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x2 + 12 x + 1)2 − 54 x2, and it has thus the irrational divisors x2 + 12 (1 + √5)x + 1, x2 + 12 (1 − √5)x + 1; and these, if we adjoin the radical √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 finding radicals by the adjunction of which the equation becomes reducible; for instance, the general quadric equation x2 + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical √(14 p2 − q).
The fundamental theorem is the Proposition I. of the “Mémoire sur les conditions de résolubilité des équations 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 substitutions 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 substitutions of the group, but further, one of which the value is invariable by these substitutions: for instance, if the equation be φ(x) = 0, then φ(x) is a function of the roots invariable by any substitution whatever. And in saying that a function is rationally known, it is meant that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.
For instance in 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 xn−1 . . . + x + 1 = 0, n a prime number, a, b, c . . . k = r , r g, r g2 . . . r gn−2, where g is a prime root of n, then the group is the cyclical group abc . . . k, bc . . . 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 first link of a long chain of questions relating to the transformation and classification of irrationals.
Returning to the question of solution by radicals, it will be readily understood that by the adjunction of a radical the group may be diminished; for instance, in the case of the general cubic, where the group is that of the six permutations, by the adjunction of the square root which enters into the solution, the group is reduced to abc, bca, cab; that is, it becomes possible to express rationally, in terms of the coefficients and of the adjoint square root, any function such as a2b + b2c + c2a which is not altered by the cyclical substitution a into b, b into c, c into a. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation.
The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained—in the first instance in the form (scarcely intelligible without further explanation) that every function of the roots x1, x2 . . . xn, invariable by the substitutions xak + b for xk, must be rationally known; and then in the equivalent form that the resolvent equation of the order 1.2. . . (n − 2) must have a rational root. In particular, the condition in order that a quintic equation may be solvable is that Lagrange’s resolvent of the order 6 may have a rational factor, a result obtained from a direct investigation in a valuable memoir by E. Luther, Crelle, t. xxxiv. (1847).
Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.
The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti (1852). See also J. A. Serret’s Cours d’algèbre supérieure, 2nd ed. (1854); 4th ed. (1877–1878).
25. Returning to quintic equations, George Birch Jerrard (1835) established the theorem that the general quintic equation is by the extraction of only square and cubic roots reducible to the form x5 + ax + b = 0, or what is the same thing, to x5 + x + b = 0. The actual reduction by means of Tschirnhausen’s theorem was effected by Charles Hermite in connexion with his elliptic-function solution of the quintic equation (1858) in a very elegant manner. It was shown by Sir James Cockle and Robert Harley (1858–1859) in connexion with the Jerrardian form, and by Arthur Cayley (1861), that Lagrange’s resolvent equation of the sixth order can be replaced by a more simple sextic equation occupying a like place in the theory.
The theory of the modular equations, more particularly for the case n = 5, has been studied by C. Hermite, L. Kronecker and F. Brioschi. In the case n = 5, the modular equation of the order 6
