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

NORMAL DOMAINS 169
In -the general equation f(x)= 0 no relation is assumed to
exist between the roots; that is, the roots are taken to be
independent variables.
In all cases a symmetric function in Q of the roots equals
a number in 2 (~ 70). If it be granted that, for the general
equation, this is the only function in 1S having this property,
condition A of ~ 155 demands simply that
Every symmetric function of the roots shall admit the substitutions of the symmetric group,
and condition B demands that
Every such symmetric function shall equal some number in 2.
Both statements are true. Hence the symmetric group is
the Galois group of the general equation.
159. Actual Determination of the Galois Group. In Exs. 1 and
4 of ~ 151 we determined the Galois groups of easy equations,
for the domain defined by the coefficients of the equation, by
the aid of the roots of the equations. When the roots are not
known, P might be obtained by the construction of the Galois
resolvent, from which Pis obtainable. But the Galois resolvent
is not easily constructed. Practically the Galois group can be
ascertained more readily from the theorem about to be deduced.
It is well to remember that, when f(x) = 0 is irreducible, the
degree of the Galois group is equal to the degree of the equation. When f(x) = 0 is reducible and the factors are known,
it is easiest to consider the equations resulting from the irreducible factors of f(x). We proceed to prove the following
theorem, in which M is any function in 0 of the roots a,.., a(,_,
which belongs to Q as a sub-group of the symmetric group:
If a function M is a number in Q, the Galois group for the
donmain 2 is either Q or one of its sub-groups.
Since, by hypothesis, M is a function in Q of the roots
a, (1l,... anl_, which is a number in 2, it follows by ~ 153 that