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

NORMAL DOMAINS

153

This means that zN is a rational function of p; that is, any
number in t12(a,,...) is a rational function of p, and lies, therefore, in the domain t(p). Conversely, any number in i(p) lies
in 0(, y...) since every number in D2(p) is a rational function of
p, and, therefore, of cc, /, y, **. This shows that Q(p) and
Q(a,a,,..) are coextensive domains, and the adjunction of a, 3,
y, ** to 0 may be replaced by the adjunction of p.
Ex. 1. Go over the above proof for the special case where
( = -vi, _ = 6/o, a = b = 1, V = 3 V/2 /5.
Here f(x) x2 2 = 0, f2(x) =x3-5= 0. Then p = 2 + /5.
There are six different p's, and II is of the sixth degree in t. Of what
degree is III?
G(t)= N(t - pl)(t - P2)... (t - P5) + il(t - p)(t - P2). (t - p5) +...
G(p) = N(p - pI)(P-P2) -.. (P - P) = 540 p2 + 360, where
p = V + /5, P3 = -2 - +V,
p2 = V2 + Co / 5, p = - + 5
P2 =V\2+c02 V5, p5 =V +2 -2V5.
By Ex. 14, ~ 71, the equation whose roots are p, pi, -, p5, is
F(t) =t6- 6t4 — 10 t3 12t2- 60t+17 = 0.
F. '(p)0 p5 - 24 p - 30 p2+ 24 p - 60.
We see that G(p) - F'(p) = 1X.
Ex. 2. Is the adjunction of V/- 2 to Q(1) equivalent to the adjunction of i + /2?
Ex. 3. Are the two domains U(j, V-, /3) and t(1, Vi) coextensive? If
not, is one a divisor of the other?
143. The Galois Domain. If f(x) = 0 is an equation of the
nth degree with distinct roots a, al, *.., (a,,, then the domain
Q(a, a...a ), obtained by the adjunction of all its roots to 12,
is called the Gclois domain of the equation f(x) = 0. Thus
the roots of the cubic x3+3x- -2x- -6-0 are -3, ~/2;
hence its Galois domain is Q2o,,-).
Ex. 1. Find the Galois domain of x4 + 6 x2 + 5 = 0.
Ex. 2. Find the Galois domain of the equation in Ex. 5, ~ 133. Show
that, in this case, (, a(,... a,,l) - (a) () 2- a ( a ) = ((a2 ) = (a