2 FUNDAMENTAL PROPERTIES OF POLYNOMIALS Booh I 
devices. The classical relations between the roots and coeffi 
cients are given by the formulae 
(4) 
l = U , l,J=U 
i= 1 u i 9 j = l u 
Each of these symmetric functions is rational in the coefficients. 
The degree of the numerator or denominator in the a’s is the 
degree of the corresponding polynomial looked upon as a func 
tion of any chosen individual root, the sum of the subscripts in 
each term of the numerator is the degree on the left in all the 
roots together. If we add symmetric polynomials of the same 
degree, or multiply two of the same or different degrees, we get 
rational fractions in the a’s which have these same properties. 
From such considerations it is easy to prove.* 
The Fundamental Theorem of Symmetric Functions 2] Any 
symmetric homogeneous polynomial in the roots of a polynomial 
in one variable is a rational fraction in the coefficients, whose 
denominator is a 0 raised to a power equal to the degree of the poly 
nomial in any one root. The numerator is homogeneous to the 
same degree, and the sum of the subscripts in each term is equal 
to the total degree of the polynomial in all the roots together. 
The degree of the numerator or denominator is called the 
‘degree’ of the symmetric function, the sum of the subscripts 
in each term its ‘weight’. A polynomial like this, where the 
sum is the same in each term is said to be isobaric. 
Definition. Any constant multiple, not 0, of the numerator 
of the fraction in the coefficients which is equal to the left-hand 
side of (3) is called the ‘discriminant’ of the polynomial. 
Theorem 3] The discriminant of the general polynomial of 
order n is of order 2(w—1) and weight n{n— 1). 
Suppose that we allow the ratios of the coefficients of our 
polynomial to vary, the roots will vary also, for if two poly 
nomials have the same roots their coefficients are proportional 
by (4). Let us suppose in particular that each coefficient takes 
a small increment. We write 
f{x) = a Q x n +a 1 x n - 1 +...-\-a n _ x x-\-a n 
cf)[x) = (a 0 +Aa 0 );r n +(&i+Aa 1 )a: n ~ 1 -{-...+ 
+ ( a M-i+Aa n _i)æ+ {a n -\- Aa n ). 
* Cf. Bôcher, p. 243. 
Chap. I I 
Let us reduce 
of /. This is dc 
Taylor’s theorei 
(f>{x+a i) = <£K) 
<f>M = K+ 
= Aa 0 Q 
The product of 
infinitesimal sin 
finitesimal, whil 
(f){x-\-oc 1 ) must 1 
one root infinite 
a Q x n -\-a x x n ' 
(a Q +l±a Q )x n +{a 
Comparing cc 
a a = a Q 
di 
d\ 
Hence d x differ; 
the same way tl 
spending c, or j 
cf>, and so on. 
In this devel 
now assume 
«( 
We may assu 
we could divide 
no vanishing ra 
fii x ) = xn f{~j 1 
(f )l {x) = X n cJ>^j