518 
EQUATION. 
[786 
is a given function M 0 + M x (o + ... + 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 &>, 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 w T as 
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 Jacobi (1837); see the memoir “ Ueber die 
Kreistheilung, u.s.w.,” Grelle, t. xxx. (1846). 
The equation x n ~ l + ...+« + 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 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, 6x, 6-x,..., 6 n ~ 1 x (6x being a rational function of x, 
and 6 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 
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, 6x, 9 x x being 
any two of the roots, we have 60 x x = Qjdx, the equation will be solvable algebraically. 
It is proper to refer also to Abel’s definition of an irreducible equation:—an equation 
cpx = 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 (f)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. 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 « 4 + « 3 + « 2 + « + l = 0; the left-hand side has here no rational 
divisor, and the equation is irreducible; but this function is = (x 2 + \x +1) 2 — -J« 2 , and 
it has thus the irrational divisors « 2 + | (1 +Jo)x+ 1, « 2 + £ (1 -Jb)x + 1; and these, 
if we adjoin the radical Jh, 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