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, [theta]x, [theta]1x being any two of the roots, we have [theta][theta]1x = [theta]1[theta]x, the equation will be solvable algebraically. It is proper to refer also to Abel's definition of an _irreducible_ equation:--an equation [phi]x = 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 [phi]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. 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 x^4 + x^3 + x^2 + x + 1 = 0; the left hand side has here no rational divisor, and the equation is irreducible; but this function is = (x^2 + 1/2 x + 1)^2 -(5/4)x^2, and it has thus the irrational divisors x^2 + 1/2(1 + [root]5)x + 1, x^2 + 1/2(1 - [root]5)x + 1; and these, if we _adjoin_ the radical [root]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 x^2 + px + q = 0 is irreducible, but it becomes reducible, breaking up into rational linear factors, when we adjoin the radical [root](1/4 p^2 - q).