Linear Algebra, Theory And Applications, 2012a

446 FIELDS AND FIELD EXTENSIONS
Then the following result is the fundamental theorem in the subject. It is the symmetric
polynomial theorem. It says that these elementary symmetric polynomials are a lot like a
basis for the symmetric polynomials.
Theorem F.1.3 Let g (x 1 ,x 2 , ··· ,x n ) be a symmetric polynomial. Then g (x 1 ,x 2 , ··· ,x n )
equals a polynomial in the elementary symmetric functions.
g (x 1 ,x 2 , ··· ,x n )= ∑ k
a k s k 1
1 ···sk n
n
and the a k are unique.
Proof: If n =1, it is obviously true because s 1 = x 1 . Suppose the theorem is true for
n − 1andg (x 1 ,x 2 , ··· ,x n ) has degree d. Let
By induction, there are unique a k such that
g ′ (x 1 ,x 2 , ··· ,x n−1 ) ≡ g (x 1 ,x 2 , ··· ,x n−1 , 0)
g ′ (x 1 ,x 2 , ··· ,x n−1 )= ∑ k
a k s ′k 1
1 ···s ′k n−1
n−1
where s ′ i is the corresponding symmetric polynomial which pertains to x 1,x 2 , ··· ,x n−1 .
Note that
s k (x 1 ,x 2 , ··· ,x n−1 , 0) = s ′ k (x 1 ,x 2 , ··· ,x n−1 )
Now consider
g (x 1 ,x 2 , ··· ,x n ) − ∑ k
a k s k 1
1 ···sk n−1
n−1 ≡ q (x 1,x 2 , ··· ,x n )
is a symmetric polynomial and it equals 0 when x n equals 0. Since it is symmetric, it is also
0 whenever x i = 0. Therefore,
q (x 1 ,x 2 , ··· ,x n )=s n h (x 1 ,x 2 , ··· ,x n )
and it follows that h (x 1 ,x 2 , ··· ,x n ) is symmetric of degree no more than d − n and is
uniquely determined. Thus, if g (x 1 ,x 2 , ··· ,x n ) is symmetric of degree d,
g (x 1 ,x 2 , ··· ,x n )= ∑ k
a k s k 1
1 ···sk n−1
n−1 + s nh (x 1 ,x 2 , ··· ,x n )
where h hasdegreenomorethand − n. Now apply the same argument to h (x 1 ,x 2 , ··· ,x n )
and continue, repeatedly obtaining a sequence of symmetric polynomials h i , of strictly decreasing
degree, obtaining expressions of the form
g (x 1 ,x 2 , ··· ,x n )= ∑ k
b k s k 1
1 ···sk n−1
n−1 skn n + s n h m (x 1 ,x 2 , ··· ,x n )
Eventually h m must be a constant or zero. By induction, each step in the argument yields
uniqueness and so, the final sum of combinations of elementary symmetric functions is
uniquely determined.
Here is a very interesting result which I saw claimed in a paper by Steinberg and Redheffer
on Lindemannn’s theorem which follows from the above corollary.