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

GALOIS THEORY OF ALGEBRAIC NUMBERS -147

where 2 N, N,... i^_ are conjugates to the primitive number
V. We now construct a new function, ~(x), as follows:
(x).' ) + N+ x)..,.
v - N X - N, x- ln-1
This is a function of x of the (i - 1)th degree,
Since N = b(a), N — = (t,,...)
and V1 = (), i(),...,
it follows that an interchange of, say, a and (ta interchanges not
only N and iVN, but also N' and N'1, and also the first two
fractions in the expression for f(x).
But <(x) is not affected by such an interchange. Hence f(x)
is not affected, no matter what two a's replace each other.
From this it follows that ~(x) is a symmetric function of
a, ar,, ***, a,,- in and the coefficients of 1(x) are numbers in Q.
If now we put x = - then 1(NL) -= 0. As N is primitive
and consequently different from N1, iVS, *, it follows that each
fraction in l(x), except the first, is zero when x -= N; for, it has
a numerator that is zero and a denominator that is finite.
The first fraction gives us -. By ~ 20 we have, for this indeterminate, the relation -'(N)-_ - N'(LN), where Q' means the
differential coefficient of q4 with respect to x. This relation
yields (1VN) = N'$'(N) or NV' = i/(N)/Q'(N7), where Q4'(IV) is
not zero, because 4P(x) has no multiple roots. Since ~(i~) and
V'(-n) are both functions of Nin 0, it follows that any number
N' can be expressed as a function in 2 of any primitive
number N.
Ex. 1. Prove that the domain t(() is identical with the domain 2(O),
N being primitive in Q(+).
Ex. 2. It was shown in Ex. 2, ~ 135, that N=X +l is a primitive
~2
number of (l, /2), where:+ V2 are the roots of the irreducible equation
x- 2 = 0. Express 5 + 3 x2, 5 and /V2, as functions of IV in (1).
Ex. 3. Express 5, i, x/2 in Ex. 2, ~ 136, each as a function in t2 of a.