THE GALOISIAN RESOLVENT AND GROUP.
Construction of n I-valued Functions; In determinants.— Let f (x)=--asxn-1- . . . +an=0 be an equation whose solution is to be effected. The first thing to do is to choose a domain of rationality R. As already remarked, the nature of R depends partly upon (1) and partly upon our own pleasure. In any case it must contain the coefficients.
Without loss of generality we may suppose its roots -unequal. For by nieans of the greatest common divisor of f(x) and f' (x) we may ob tain by rational operations an equation whose roots are the distinct roots of (1).
Let us now adjoin n new variables u,, ...
to R, forming a domain R', and introduce the rational function iti.r1-1- . . . unxn. (4) If we permute the xi, x,, . . . xn in all pos sible ways, or, as we say, apply the n ! substitu tion (x, x, XX.
xg, . • . of the symmetric group, we get the n! functions V,, V1, . . . Vn!. (5) With these we form the equation P(t;u,, mil =(t 0— . Tint1-0. (6) whose coefficients lie in R' In the nant of (6), D(u,,... un), we may give to 141, un values, at , . . . in R, integral values, even, if we choose, in an infinity of ways so that • 0. In that case the quantics (5) are dis tinct and the roots of (6) thus unequal. The function (4) has thus n ! values under the sym metric group. A special case of this function (4) was used by Lagrange; in its general form it was first employed by Abel. Its fundamental importance in the solution of algebraic equa tions was first brought out by Galois. For this reason the function V in (4) is called the Galoisian resolvent function. Besides the func tion (4) there are obviously an infinity of other n l-valued functions. The function (4) is em ployed on account of its simplicity.
On replacing the u's by the a 's these variables disappear. Their introduction was to show the existence of n I-valued rational functions of the roots xi, ... xn. Such auxiliary variables which we introduce into our reasoning, and which at any moment can be made to disappear by giving them appropriate special values, are called in determinate:. In a primitive way they are used by all mathematicians. Kronecker has shown that they are an implement of immense power in algebraic investigations. Since in the end we can always replace the indeterminates by values lying in our domain, we shall suppose that our domain contains in advance as many of these auxiliary variables as we care to use.
Galoielan Resolvent and In gen eral the equation (6) is reducible in R, so that F(t).--Ge (t, tti, un) (t, . . . un) . . . Let us take now any one of these irreducible factors, say that one which admits V, as root, to form the equation Go (t, ul, un) 0. (7) This is called the Galoisian Resolvent of (1) for the domain R. Let its degree in t be Rt.
Galois showed now that the solutions of (1) and (7) are equivalent problems. In fact every rational function of the roots of (1), and in particular the roots themselves and hence also the roots Vs, Vs, . . . Via of (7), are ra tional functions of V,. We have therefore for any rational function of the x's 0(xi, xn).04-r, . . .± rgyr-1 The advance that is made by considering the equation (7) instead of the original equation (I) lies in the fact that the roots of (7) are rational in any one of them. Let the roots of (7) be Vf, . .
These are obtained from the expression (4) by effecting certain substitutions, = 1, S2, . on the roots xi, . . . xn. These m substitutions G enjoy now three remarkable properties: 1° Every rational function xn) of the roots of (1) which remains una'tered by G lies in R, or, as we say, is rationa'ly known.
2° If the rational function of the roots xn) is rationally known, it remains un altered for the substitutions G.
3° The substitutions G form a group, and there is no other group of substitutions having the properties 1°, 2°.
This group is called the Galoisian group of the equation (1) for the domain R. For the definition of the various terms concerning groups see GROUPS, THEORY or. The index of a sub-group H of a group G with respect to G is the ratio between the number. of terms in H and the number of terms in G. We say for the domain R, because by changing R the irre ducible factors of (6) will in general change, and therefore the substitutions G will in gen eral change. The importance of the Galoisian group, or, as we shall say more shortly, the Group, of an equation f(x) =0 lies in the fact that an investigation of its structure reveals many of the most important properties of the algebraic irrationalities defined by this equation. In particular it affords a rational and uniform scheme for effecting the solution of any alge braic equation. Before entering on this topic let us consider