Theorem 1
Let

be a symmetric polynomial (with integer coefficients)
in

. Then

can be expressed as a polynomial
(with integer coefficients) in
the elementary symmetric functions

given
by
Proof.
The idea is to deal with the terms of

``from the outside in''. That is,
we first deal with terms which are as ``unbalanced'' as possible.
We can write
We put an ordering on

-tuples by saying that

if and only if

.
Now sort the terms in decreasing order by

.
Choose the biggest term

and notice that the polynomial
has the same largest term, and the coefficient of that term is 1.
So subtract off

times this product, and repeat.