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

158

THEORY OF EQUATIONS

where p is a root of the resolvent (~ 144), and are, therefore,
functions in t2 of the one root p (~ 138).
To prove the second part, let f(x)= 0 be a normal equation,
having the roots a, Ta, _., a,-_. Then Q() is a normal domain
(~ 141); f()= 0 is its own Galois resolvent, because being
irreducible it satisfies property (1) in ~ 145, and all its roots
being in the domain t2(a), and, therefore, functions of a in Ut, it
satisfies also properties (2) and (3).
Ex. 1. Show that the equation in Ex. 5 (~ 133) is its own Galois
resolvent.
Ex. 2. Show that the Galois resolvent in Ex. 2 (~ 145) satisfies the
definition of a normal equation.
Ex. 3. Find the Galois domain for the equation in Ex. 3 (~ 133).
Find the irreducible equation in 0(1) having the primitive number 6 + V/5
as a root. Show that this equation is its own Galois resolvent and that
the Galois domain is normal.
147. Theorem. f f(x) = 0 is a normal equation of the nth
degree with a root p as a primitive number in the normal domain
f2,p) then the transposition (pp.,) causes each of the numbers
conjugate to p to be replaced by some other of their own set, but no
two numbers are replaced by the same one.
Let the numbers conjugate to p be p, pi, *-, p,-l. They are
all roots of the equation f(x)=0 (~ 138). Since Q(p) is
assumed to be normal, they are contained in it. Hence we
have p = o(p), pi = li(p),.*., p, - -= C,-1(P), I
where 40, bl,.. are functions in Q. If in 1k(p), which is a root
of f(x) = 0, we replace p by p,, we get as a result 'b(Ph, which,
being conjugate to )+(p), is another root of f(x) =0 (~ 136).
Hence the numbers in the series
0(ph), ql(Ph)0 **',,n-i(Ph) II
are identical with numbers in I, except in the order in which
they are written. Now, if we can show that the roots II are
all distinct, our theorem is proved.