The theory is an extensive and important one, depending on the notions of substitutions and of groups (q.v.).
15. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.g. in the case of a quartic equation, the function ab + cd, it is in general possible to determine rationally the value of any similar function, such as (a + b)³ + (c + d)³.
The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose that the roots of a quartic equation are 1, 2, 3, 4, then if it is given that ab + cd = 14, this in effect determines a, b to be 1, 2 and c, d to be 3, 4 (viz. a = 1, b = 2 or a = 2, b = 1, and c = 3, d = 4 or c = 3, d = 4) or else a, b to be 3, 4 and c, d to be 1, 2; and it therefore in effect determines (a + b)³ + (c + d)³ to be = 370, and not any other value; that is, (a + b)³ + (c + d)³, as having a single value, must be determinable rationally. And we can in the same way account for cases of failure as regards particular equations; thus, the roots being 1, 2, 3, 4 as before, a²b = 2 determines a to be = 1 and b to be = 2, but if the roots had been 1, 2, 4, 16 then a²b = 16 does not uniquely determine a, b but only makes them to be 1, 16 or 2, 4 respectively.
As to the a posteriori proof, assume, for instance,
t1 = ab + cd, y1 = (a + b)³ + (c + d)³,
t2 = ac + bd, y2 = (a + c)³ + (b + d)³,
t3 = ad + bc, y3 = (a + d)³ + (b + c)³;
then y1 + y2 + y3, t1y1 + t2y2 + t3y3, t1²y1 + t2²y2 + t3²y3 will be respectively symmetrical functions of the roots of the quartic, and therefore rational and integral functions of the coefficients; that is, they will be known.
Suppose for a moment that t1, t2, t3 are all known; then the equations being linear in y1, y2, y3 these can be expressed rationally in terms of the coefficients and of t1, t2, t3; that is, y1, y2, y3 will be known. But observe further that y1 is obtained as a function of t1, t2, t3 symmetrical as regards t2, t3; it can therefore be expressed as a rational function of t1 and of t2 + t3, t2t3, and thence as a rational function of t1 and of t1 + t2 + t3, t1t2 + t1t3 + t2t3, t1t2t3; but these last are symmetrical functions of the roots, and as such they are expressible rationally in terms of the coefficients; that is, y1 will be expressed as a rational function of t1 and of the coefficients; or t1 (alone, not t2 or t3) being known, y1 will be rationally determined.
16. We now consider the question of the algebraical solution of equations, or, more accurately, that of the solution of equations by radicals.
In the case of a quadric equation x² − px + q = 0, we can by the assistance of the sign √( ) or ( )1/2 find an expression for x as a 2-valued function of the coefficients p, q such that substituting this value in the equation, the equation is thereby identically satisfied; it has been found that this expression is
x = ½ {p ± √(p² − 4q) },