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

GALOIS THEORY OF ALGEBRAIC NUMBERS 149

is of the nth degree and 0(x) = 0 of the nith degree, n must be
a multiple of n,, that is, n = n1n2.
Ex. 1. As an illustration, take the irreducible equation f(x)-x4+- 10.
It has the roots a = — V2( I + i), cl -- V/2(1 +), = 2( — /( i),
C3 -— 2 v'2 (1 —i). Let = 0 (t) Ca2, then NA- l = i and N2 =, - - -i.
Hence, ( (x) ( - (x 4- i)2( 2 = (x' + 1)2 = 0. We have 0(x)=x2 + 1 = 0,
which is satisfied by x, T, AT,.. The equation 0[) (:C ) ] =() (z2)'2 +1 =is satisfied by c, ixl, Uc, ic3, the roots of f() = 0.
Ex. 2. From the roots of the equation in Ex. 5, ~ 133, find JV, IT1, 2s, N3,
when AN a" + 4 3. Determilne whether tlie equation cp(x) = 0 is in this
case reducible; if it is, find n1 and 1n. and show that 0[0(a)] - 0 is satisfied by the roots of the given equation /(x) = 0.
Ex. 3. From the roots of the equation in Ex. 5, ~ 133, find l,, N2, N3,
when N = 4 a. Is c(x) = O reducible?
Ex. 4. In Ex. 5, ~ 135, form 4(y) = 0 and examine its reducibility,
when N= C-2.
139. Normal Equations. A normal equation is an irreducible
equation in which each root can be expressed as a function in
2 of one of the roots.
Ex. 1. The roots (x1, (X2, (C3, of x4 +1 == 0, Ex. 1, ~ 138, may be expressed
in terms of a thus: a = - -, = -- aX3, = -- c3. Hence x4 + 1 = 0,
being irreducible, is normal.
Ex. 2. Show that x4 + xc3 + x2 + x + 1 = 0 is a normal equation.
Ex. 3. Show that x' - 2 x2 + 9 = 0 is normal.