SOME P1ROPERTIES OF THE GALOISIAN GROUP G.
Since the group G of an equation (tort -1-atxn-1 + ... +an= 0 (I) is unique for a given domain of rationality R, it follows; 1° that the group is independent of the particular ts I-valued function we take; 2° that we get the same group whichever of the irredncible factors Go(t) . . of (6) we may choose; and 3° th'at these functions Go, G., . . . are all of the same degree.
4°. /n any rational equation 0(xl, xu) betweevt the roots of (1) the substitutions of G may be aPPlied, and the result is a true equation.
This is not true for all substitutions. For example, let f (x) x1-1 whose roots are 2 irim e . M=0, 1, 2.
Take as domain R(1), and as rational rela tion XiX2 = On applying the substitution 1, 2) xixoco this relation becomes, xo.ro=1, which is false.
Group Belonging to a Rational Function of the Roots and Rational Functions Belong ing to a Group.—Let 0(xi, . . x.) be a rational function of the roots of (1). Since the group G of (1) contains the identical sub titution, ye remains unaltered by at least one substitution of G and may remain unaltered by others. These substitutions form a subgroup
of G called the group belonging to On the other hand, let H be a subgroup of G. Any rational function igzi, . . . Xn) which remains unaltered by the substitutions of H but is changed by all other substitutions of G is said to belong to H. It is important to note that while the substitutions of the Galoisian group which leave a rational function . . . xn) unaltered form a group, this property does not hold for substitutions which lie outside G. For example, the substitutions of the symmetric group So which leave 2-ins 40-=xix•, , 1, 2, 3, 4, 5, do not form a group. This is due to the fact that the group of the equation x°— 1=3, the domain being R(1), is not but a smaller group.
If a(x., . . xn), . . xn) belcmg to the same subgroup H of the Galoisian grout, each can be expressed rationally in terms af the other.