## 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:

\begin{eqnarray*} e_1(x_1, \ldots , x_n) & = & x_1 + x_2 + \cdots + x_n \\ e_2(x_1, \ldots , x_n) & = & x_1x_2 + x_1x_3 + \cdots + x_1x_n + x_2x_3 + \cdots + x_{n-1}x_n \\ & \vdots & \\ e_i(x_1, \ldots , x_n) & = & \sum_{1\leq j_1 < j_2 < \cdots < j_i \leq n} x_{j_1} \dotsm x_{j_i} \\ & \vdots & \\ e_n(x_1, \ldots , x_n) & = & x_1x_2 \dotsm x_n \end{eqnarray*}

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 Gauss2, which will here be more precisely stated and proved.

© 2016 George S. of www.notionsandnotes.org.