RATIONAL RFSOLVENTS.
Let iP(xo, xn) be a rational function of t.he roots of ttexn-Fal-1- +an= 0, (I) whose group for the domain R is G. Let 0 belong to a subgroup H of G of index r. Then on applying the substitutions of G to a it will take on r distinct values, • • • 01.--1, (8) which are called conjugate functions. They are in fact roots of an irreducible equation 4)(Y) = CY — 4') — 91) • • . — 1), (9) whose coefficients lie in R. 1t is thus a rational equation. Suppose one of its roots, say can be found. If we adjoin it to R, forming a domain R', the group of (1) is no longer G, but H.
Suppose not only cti but all the roots of (9) can be found. Their adjunction to R forms a domain R" for which the group of (1). is the greatest invariant subgroup of G contained in H. In any case the adjunction of one or more roots of (9) produces a reduction of the group of the given equation (1). But in reducing the group of this equation we have made a step in its solution. For when the domain of rationality has been enlarged to such an extent that the group of the equation (1) embraces only the identical substitution, the roots of (1) are rationally known, that is, can be expressed rationally in terms of quantities lying in that domain of rationality. The equation (9) is
called a resolvent equation, or more specifically a rational resolvent, since its roots 0, . . . are rational functions of the roots of a given equation (1).
The group of the resolvent equation (9) is of importance sometimes. In the functions (8) considered as functions of the x's, let us effect the substitutions of the group G. This gives rise to a substitution group I' in the ites, and this group is the group of the resolvent equa tion (9), the domain of rationality being that of G, viz., R. The groups G and f are what is called meroedrically isomorphic. To the identical substitution of f corresponds the group / above mentioned, viz., the subgroup of G, which leaves all the roots (8) unaltered. To any subgroup I', of f will correspond a subgroup G. of G, and conversely. In particular if f. is an invariant subgroup, G. is also invariant.