i.e. here a³, b³ are each of them a 2-valued function, but as the only effect of altering the sign of the quadric radical is to interchange a³, b³, they may be regarded as each of them 1-valued; a and b are each of them 3-valued (for observe that here only a³b³, not ab, is given); and ab(a + b) thus is in appearance a 9-valued function; but it can easily be shown that it is (as it ought to be) only 3-valued.
In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression
x = 3√[½ {r + √(r² − 4⁄27 q³) }] + 1⁄3 q ÷ 3√[½ {r + √(r² − 4⁄27 q³) }],
this may depend on an expression of the form 3√(γ + δi) where γ and δ are real numbers (it will do so if r² − 4⁄27 q³ is a negative number), and then we cannot by the extraction of any root or roots of real positive numbers reduce 3√(γ + δi) to the form c + di, c and d real numbers; hence here the algebraical solution does not give the numerical solution, and we have here the so-called “irreducible case” of a cubic equation. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.
The irreducible case is solvable by a trigonometrical formula, but this is not a solution by radicals: it consists in effect in reducing the given numerical cubic (not to a cubic of the form z³ = a, solvable by the extraction of a cube root, but) to a cubic of the form 4x³ − 3x = a, corresponding to the equation 4 cos³ θ − 3 cos θ = cos 3θ which serves to determine cosθ when cos 3θ is known. The theory is applicable to an algebraical cubic equation; say that such an equation, if it can be reduced to the form 4x³ − 3x = a, is solvable by “trisection”—then the general cubic equation is solvable by trisection.
18. A quartic equation is solvable by radicals, and it is to be remarked that the existence of such a solution depends on the existence of 3-valued functions such as ab + cd of the four roots (a, b, c, d): by what precedes ab + cd is the root of a cubic equation, which equation is solvable by radicals: hence ab + cd can be found by radicals; and since abcd is a given function, ab and cd can then be found by radicals. But by what precedes, if ab be known then any similar function, say a + b, is obtainable rationally; and then from the values of a + b and ab we may by radicals obtain the value of a or b, that is, an expression for the root of the given quartic equation: the expression ultimately obtained is 4-valued, corresponding to the different values of the several radicals which enter therein, and we have thus the expression by radicals of each of the four roots of the quartic equation. But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one.
It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of k-valued functions of the five roots (a, b, c, d, e); the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N.H. Abel (1824) showed that a general quintic equation is not solvable by radicals; and a fortiori the general equation of any order higher than 5 is not solvable by radicals.
19. The general theory of the solvability of an equation by radicals depends fundamentally on A.T. Vandermonde’s remark (1770) that, supposing an equation is solvable by radicals, and that we have therefore an algebraical expression of x in terms of the coefficients, then substituting for the coefficients their values in terms of the roots, the resulting expression must reduce itself to any one at pleasure of the roots a, b, c ...; thus in the case of the quadric equation, in the expression x = ½ {p + √(p² − 4q) }, substituting for p and q their values, and observing that (a + b)² − 4ab = (a − b)², this becomes x = ½ {a + b + √(a − b)²}, the value being a or b according as the radical is taken to be +(a − b) or −(a − b).
So in the cubic equation x³ − px² + qx − r = 0, if the roots are a, b, c, and if ω is used to denote an imaginary cube root of unity, ω² + ω + 1 = 0, then writing for shortness p = a + b + c, L = a + ωb + ω²c, M = a + ω²b + ωc, it is at once seen that LM, L³ + M³, and therefore also (L³ − M³)² are symmetrical functions of the roots, and consequently rational functions of the coefficients; hence
½ {L³ + M³ + √(L³ − M³)²}