Linear Algebra, Theory And Applications, 2012a

F.2. THE FUNDAMENTAL THEOREM OF ALGEBRA 447
Theorem F.1.4 Let α 1 , ··· ,α n be roots of the polynomial equation
a n x n + a n−1 x n−1 + ···+ a 1 x + a 0 =0
where each a i is an integer. Then any symmetric polynomial in the quantities a n α 1 , ··· ,a n α n
having integer coefficients is also an integer. Also any symmetric polynomial in the quantities
α 1 , ··· ,α n having rational coefficients is a rational number.
Proof: Let f (x 1 , ··· ,x n ) be the symmetric polynomial. Thus
f (x 1 , ··· ,x n ) ∈ Z [x 1 ···x n ]
From Corollary F.1.3 it follows there are integers a k1···k n
such that
∑
f (x 1 , ··· ,x n )=
a k1···k n
p k 1
1 ···pk n
n
k 1 +···+k n ≤m
where the p i are the elementary symmetric polynomials defined as the coefficients of
Thus
n∏
(x − x j )
j=1
=
f (a n α 1 , ··· ,a n α n )
∑
a k1···k n
p k1
1 (a nα 1 , ··· ,a n α n ) ···p k n
n (a n α 1 , ··· ,a n α n )
k 1+···+k n
Now the given polynomial is of the form
∏
n
a n
j=1
(x − α j )
and so the coefficient of x n−k is p k (α 1 , ··· ,α n ) a n = a n−k . Also
p k (a n α 1 , ··· ,a n α n )=a k np k (α 1 , ··· ,α n )=a k a n−k
n
a n
It follows
f (a n α 1 , ··· ,a n α n )=
∑ ( ) k1
( ) k2
(
a k1···k n
a 1 a n−1
n a 2 a n−2
n ···
a n a n
k 1 +···+k n
a n a 0
n
a n
) kn
which is an integer. To see the last claim follows from this, take the symmetric polynomial
in α 1 , ··· ,α n and multiply by the product of the denominators of the rational coefficients
to get one which has integer coefficients. Then by the first part, each homogeneous term is
just an integer divided by a n raised to some power.
F.2 The Fundamental Theorem Of Algebra
This is devoted to a mostly algebraic proof of the fundamental theorem of algebra. It
depends on the interesting results about symmetric polynomials which are presented above.
I found it on the Wikipedia article about the fundamental theorem of algebra. You google