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 = 1/2{p + [root](p^2 - 4q)}, substituting for p and q their values, and observing that (a + b)^2 - 4ab = (a - b)^2, this becomes x = 1/2{a + b + [root](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 x^3 - px^2 + qx - r = 0, if the roots are a, b, c, and if [omega] is used to denote an imaginary cube root of unity, [omega]^2 + [omega] + 1 = 0, then writing for shortness p = a + b + c, L = a + [omega]b + [omega]^2 c, M = a + [omega]^2 b + [omega]c, it is at once seen that LM, L^3 + M^3, and therefore also (L^3 - M^3)^2 are symmetrical functions of the roots, and consequently rational functions of the coefficients: hence 1/2{L^3 + M^3 + [root](L^3 - M^3)^2} is a rational function of the coefficients, which when these are replaced by their values as functions of the roots becomes, according to the sign given to the quadric radical, = L^3 or M^3; taking it = L^3, the cube root of the expression has the three values L, [omega]L, [omega]^2 L; and LM divided by the same cube root has therefore the values M, [omega]^2M, [omega]M; whence finally the expression (1/3)[p + [root 3]{1/2(L^3 + M^3 + [root](L^3 - M^3)^2)} + LM / [root 3]{1/2L^3 + M^3 + [root](L^3 - M^3)^2}] has the three values (1/3)(p + L + M), (1/3)(p + [omega]L + [omega]^2 M), (1/3)(p + [omega]^2 L + [omega]M); that is, these are = a, b, c respectively. If the value M^3 had been taken instead of L^3, then the expression would have had the same three values a, b, c. Comparing the solution given for the cubic x^3 + qx - r = 0, it will readily be seen that the two solutions are identical, and that the function r^2 - (4/27)q^3 under the radical sign must (by aid of the relation p = 0 which subsists in this case) reduce itself to (L^3 - M^3)^2; it is only by each radical being equal to a rational function of the roots that the final expression _can_ become equal to the roots a, b, c respectively.