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.
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Determinants of Matrices over Commutative Rings

Let \(R\) be a commutative ring. Fix an integer \(n \geq 1\). Let \(\mathbf{A} = (a_{ij})\) be the \(n \times n\) square matrix

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

with entry \(a_{ij}\in R\) and \(1 \leq i,j \leq n\) is at the intersection of the \(i^\mathrm{th}\) row and \(j^\mathrm{th}\) column. We will define a function \(\det(\mathbf{A}) =\det (a_{ij})\) of this matrix \(\mathbf{A}\), which will also be denoted by

$$\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}$$

We will also state and prove the various familiar properties of the determinant in this setting.
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