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

GALOIS THEOI:RY OF ALGEBRi3AIC NUMlBERS 145

More generally, if (t (i) = a - ib), where a and b are rational numbers,
then 0(- i) a - i; if 0(i) -mf- = a, then (- i) = a. Hence the imr
primitive numbers are in this example confined to those that are rational,
and the domain Q(i,i) is primitive. Since both (21,i) and Q(1,-i) are
domains containing numbers a + ib, where a and b are rational, and may
be positive or negative, it follows that the two conjugate domains are
identical. Hence (1, i) is a normal domain.
Ex. 2. The roots of the irreducible equation x2-2=0 are ~ V2. Show
that /2 + is a primitive number of 2(, V2), that 10 is imprimitive, that
V/2
the domain l(i, v,) is primitive and normal.
Ex. 3. If a is a root of x2 + 10 x + 1 = 0, define the functions of a such
that N will be the imprimitive number 5.
Ex. 4. Show that the number 1V= a2 + a3, belonging to the normal
domain (1,,), in Ex. 2, ~ 67, is imprimitive and that the domain Q(1, )
is imprimitive.
Ex. 5. If J= a2, where a is a root of x + 1 =0, show that NV is imprimitive, that NJV = a2 - a is primitive, that the domain l2(, ) is normal and
imprimitive.
Ex. 6. If N - ac6 and a is a root of X8 + 1 = 0, prove that N is irmprimitive, that 0l(, a) is normal and inlprimitive.
Ex. 7. If a is a root of x7 - 1 = 0, prove that 2(,,) is imprimitive.
136. Theorem. Every number N in the tdomcain i () of the nth
degree is the root of some equation of the nth degree in 0, the
other roots of which are the remaining numbers conjugate to N,
viz. N, N2,... * _n-1
Take the product
(y - N)(y - N)... (y - 0_,) = yn +pyn-l + +pn- (y),
in which - p = N + N1 + * * + Vn_1,
p2 = NNI + lNV, +... +,,_X-2N1_
~ 2p, = ^NN... NL.
We see that all the coefficients pI, P2, *, p, ^are rational symmetric functions of the numbers iV,, 'V,,_~ — Since N-= >(a),
~=1 = ((ci), *'" B, — N e ((n- ) (~ 134), where <q is a function in
L