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

154

THEORY OF EQUATIONS

144. Theorem. The Galois domain of any algebraic equation
is a normal domain.
The degree of the Galois domain (a a.,, _,) is not usually
the same as that of the equation f(x) = 0; let it be m.
Let p be a primitive number of the Galois domain, then
o(a, -.,a,,_) = Q(P)It follows that p is a root of an irreducible equation of the
degree m (~ 138), viz. the equation
g(y) =O, I
The root p, being a number in the Galois domain, can be
expressed as a function of a0, a1, *.., ca_, in 0; that is,
p=J;(ao, a,1,,O a,-1), II
Consider all the permutations which can be performed with
the n subscripts of the letters a, taken all at a time. The
number of these permutations is n! They correspond to the
symmetric group of substitutions (~ 98).
If we operate upon the subscripts in II with each substitution of the symmetric group of the order n!, in turn, we obtain
values for p which we indicate, respectively, by
P, PD '", Pn!-1 III
Next, if we operate with any substitution of the symmetric
group upon the p's in III, we get the same set of p's over again,
only in a different order; for, any number resulting from this
second operation is obtained from II by two substitutions, the
product of which, by definition of a group, is identical with
one of the substitutions in the group. Hence, if we form the
equation = IV
equation H(y) (y- p)(y- pi)... (- P!- 1)= 0, IV
this equation is invariant under any of the substitutions of
the symmetric group; hence, the coefficients of y, obtained by
performing the indicated multiplications in IV, are invariant.