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

SYMMETRIC FUNCTIONS OF THE ROOTS

87

70. Fundamental Theorem of Symmetric Functions. Every
rational symmetric fulnction of the roots of' an algebraic equation
can be expr essed rationally in terms of the coeficients.
To begin with, we shall find the value of the symmetric
function SoamPp, in which each term involves two of the roots.
We have
Se= i~ - m + yn + +...
S a = e + 3P + ' +.
Multiplying, we get
SSp = m+p +_ m+. + Yn+p n+.
+ m3 U nyP + /+/ nyp +...,
that is, SmSp = s,n+p + ( anfP,
hence, S(/p p= smsp - s,'+. I
This result has been obtained on the supposition that m and p are
unequal integers. If they are equal, then the terms in,oa"/nP
become equal two and two, and Sa"?nP '= 2 (a,3)m = Sm - s.2m
In either case the symmetric function is expressed as a rational
function of the sums of powers of the roots. But by ~ 68 the
sums of like powers, Sn can be expressed rationally in terms of
the coefficients of the given equation. Hence S(/"P can be expressed rationally in terms of the coefficients.
Next we express the value of the symmetric function aESPPyq,
where each term involves three roots, as a rational function of
the coefficients. We have
a1(jp3 =P (/ tp- + aylp + /pnyp +...,
s, = + /3 + Y +
Multiplying, we have
sPqapa = / am+qp3 + /n4+qy + yt+q 4p +
+ amfp+q + fi3mPp+ + ymap+q +.
+ atmp^q +....