An introduction to the modern theory of equations, by Florian Cajori.

GALOIS RESOLVENT BY ADJUNCTION

179

164. A Resolution of the Galois Resolvent. Let the Galois
resolvent g(y) = 0 have a root p. If we effect upon p tie substitutions s, of the sub-group Q, one at a time, we get the values
P, pl, P2, ", Pq-1l I
where p, is gotten by operating upon p with the substitution s,.
If upon the p's in I we effect any substitution of the group Q,
the pi in I simply undergo a permutation; for, each result thus
obtained, being derived from p by effecting two substitutions
in succession, is equivalent to p, operated upon by that substitution of Q which is the product of those two substitutions.
Hence, (y, M) (y — p) (y - pi) ' (y - p-I), II
is invariant under Q, and the coefficients of y in expression
II are numbers in S(uM), ~ 162. By the notation g(y, M) we
mean here a function of y in which the coefficients of y are
numbers in tQ(M)
Now g(yI, M) is a divisor of g(?y) in the domain Q(,), for the
former is of degree q, the latter of p, and p =jq, ~ 160.
If upon II we effect a substitution t which occurs in P, but
not in Q, we get
g(y, Y) ( - p) ( - pI), )... (y - P-_(t)). III
The values p,(t) p,(t),.., pq_-( are roots of g(y) = 0, hence III is
also a divisor of g(y).
Two sets of roots p(t), *.., p )-l(t) obtained from two distinct substitutions t, are either indentical or they have no root in common.
Consequently, two distinct functions g(y, lM) have no common
factor, and we have the resolution into distinct factors
g(y) g(y, 1) ). (g(, M1,). g(y, j_-) IV
It is to be noticed that in this resolution the factors g(y, Me) do
not usually belong to the same domain; they belong respectively
to the domains.) O(M,),,), t, (M.j. Another resolution of
g(y) is possible, in which all the factors belong to the same
domain Q(M).