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

NOR(3MAL DOIMAITNS

155

But these coefficients are functions in t2 of the roots p, pi,.,
and by relation II, also functions in Q of 0o, a(, *., c,,_l.
Because of the invariance of the coefficients of IV under the
symmetric group, they are symmetric functions in Q of ao, a,,.., a,,_i, i.e. symmetric functions in Q of the roots of f(x) =0.
Hence IV is an equation in 2 (~ 123), and its roots are numbers
in (a,..., n
But p is a root of both H(y) = 0 and g(y) = 0. Since g(y) - 0
is irreducible, all its roots must be roots of H(y)= 0 (~ 126).
But all the roots of H(y) = 0 are numbers in 2( *, a _-);
hence all the roots of g(y)= 0 (viz. the conjugate numbers
P, Pi, **, Pm-i) are numbers in 02a,, an_1 But
Q(P) -= 0(a,., an,-1)'
hence we have 2(p)= ) ( = *p) =. (Pm-P.
That is, 0(a,., an _ is a normal domain.
145. Galois Resolvent. The equation g(y) = 0 of ~ 144 is
called the Galois resolvent of the given equation f(x) = 0 in the
domain Q, defined by the coefficients of the equation f(x) = 0.
This resolvent possesses the following properties:
(1) g(y) = 0 is irreducible.
(2) Each root of f(x) = 0 can be expressed as a fiuction in 2
of one root p of the equation g(y) =0. That is, each of the
roots a,. al,..,, a-1 occurs in Q(a,...,, )? a domain equivalent
to i(p).
(3) One root p of g(y) = 0 can be expressed as a function in 2
of the n roots off(x)= 0. That is, by II, ~ 144, we have
p =fi((t, a, "', a-l)'
Ex. 1. The cubic x3 + 32 + x - 1 = O has the roots
c =- 1, lx=-1 +x/2, 2%=-l1-x/2.
Hence the Galois domain is 0(1, 2). Also, p =v/2 is a root of the
irreducible equation g(y) = x - 2 - 0 and is a primitive number of the