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

THE HOMOGRAPHIC TRANSFORMATIONS

101

Every rational function of a root am can be expressed in the
form of a fraction whose numerator and denominator are each
rational integral functions of the root, viz.
h(am)
Multiplying both numerator and denominator of - by the
same quantity, we may write
1(a-) 1h(a*,). h(a,).. *(,(,)
We see that the denominator h(a1) * h(a()... h(an) is a symmetric function of the roots a,, az "', a,, of the equation
f(x) = 0. By ~ 70 this function can be expressed rationally
in terms of the coefficients. Hence ta,, can be made to disappear from the denominator of the fraction representing the
1_ _ _ _ _ 1
value of 1( In other words, ) is reduced to an integral
h'((c.) h(am)
fanction ot' (t,.
Again, the numerator of this fraction, viz.
7h((l).....A 1_) h(~(m+1) ~*(, ( )c
is a symmetric function of the roots aC, *..,,,_,,,,,n+, *. ac, of the
equation — (x) = 0. Hence it can be expressed as a rational
X - am,
function of the coefficients of this equation. lThese coefficients
are rational integral functions of a,, and the coefficients of
f(x) = 0, as may be seen by performing the indicated division.
Hence 1 and also g(") can be expressed as an integral
lz(am) 7th(a) - (a)
rational function of at,,. Let the integral function G(c,,) —,)
If G(ax) is of a degree higher than the nth, divide G(x) by
f(x), and we obtain
G(x) = Q f(x) + 1H(x),