We are going to be dealing with
polynomials in
n variables
x_{1}, x_{2}, ..., x_{n}. Such a polynomial
is called
symmetric if any rearrangement of the variables
leaves the polynomial unchanged.
Formally, a polynomial f(x_{1},...,x_{n})
is symmetric iff
f(x_{1},...,x_{n})=f(x_{s(1)},...,x_{s(n)})
for any permutation s in the nth Symmetric group
S_{n}
For example,

If n=1, so there is just one variable, then every polynomial is symmetric.

If n=2, then
 x_{1} + x_{2} is symmetric.
 x_{1}x_{2} is symmetric.
 x_{1} + (x_{2})^{2} is not. (Because
when we swap over x_{1} and x_{2} for this
polynomial we get x_{2} + (x_{1})^{2}.)
 If n=3, then
 x_{1} + x_{2} is not symmetric because
if we consider the permutation that sends 1 > 2 > 3
and apply it to our polynomial we get x_{2} + x_{3}.
 x_{1} + x_{2} + x_{3} is symmetric
though and so is (x_{1})^{2} + (x_{2})^{2} + (x_{3})^{2}.
The elementary symmetric polynomials are defined as follows:

First we take the sum of all the variables
s_{1}=Sum(1<= i <=n) x_{i}.
Thus, s_{1}=x_{1}+x_{2} + ... + x_{n}.

Next we take the sum of all products of pairs of distinct variables
s_{2}=Sum(1<= i < j <=n) x_{i}x_{j}.
 Then we take the sum of all products of three distinct variables
s_{3}=Sum(1<= i < j < k <=n) x_{i}x_{j}x_{k}.

...

Finally we take the product of all of the variables
s_{n}=x_{1}...x_{n}
It is not too hard to see that s_{1}, ..., s_{n}
are symmetric polynomials.
In the case n=2 we get

s_{1}=x_{1} + x_{2}.
 s_{2}=x_{1}x_{2}.
For n=3 we get
 s_{1}=x_{1} + x_{2} + x_{3}.
 s_{2}=x_{1}x_{2} + x_{1}x_{3} + x_{2}x_{3}.
 s_{3}=x_{1}x_{2}x_{3}.
It turns out that any symmetric polynomial can be constructed out of these elementary
ones.
Theorem Let f(x_{1},...,x_{n}) be a symmetric
polynomial over a field then there exists a polynomial
g(x_{1},...,x_{n}) such that
f(x_{1},...,x_{n})=g(s_{1},...,s_{n}).
Here is a proof of the fundamental theorem on symmetric polynomials.