## Symmetric polynomials, fundamental theorem of symmetric functions and an application

Let \(R\) be a commutative ring and \(R[x_1, x_2 , \ldots , x_n]\) the polynomial ring in \(n\) indeterminates over \(R\). A polynomial \(f(x)\) in it is said to be a symmetric polynomial (or symmetric function) of \(x_1, \ldots , x_n\) if it is unchanged by any permutation of the indeterminates. Consider the polynomials:

All of \(f(t ,x_1, \ldots, x_n)\) and \(e_i(x_1, \ldots , x_n)\) are symmetric functions of \(x_1, \ldots , x_n\).
The \(e_i\)'s are called elementary symmetric polynomials of \(x_1, \ldots , x_n\).
Sums, products and scalar multiples of symmetric polynomials are again symmetric polynomials.
A polynomial \(g(e_1, \ldots , e_n)\) of \(e_1, \ldots , e_n\) becomes a symmetric polynomial \(g(x_1, \ldots x_n)\) when \(e_i\)'s are written in terms of \(x_i\)'s.
The fundamental theorem on symmetric functions is the converse of this statement, credited to Gauss^{2},
which will here be more precisely stated and proved.

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