fundamental theorem of symmetric polynomials

Every symmetric polynomial  $P(x_1,x_2,\mathrm{\dots },x_n)$  in the indeterminates $x_1,x_2,\mathrm{\dots },x_n$ can be expressed as a polynomial  $Q(p_1,p_2,\mathrm{\dots },p_n)$  in the elementary symmetric polynomials$p_1,p_2,\mathrm{\dots },p_n$ of $x_1,x_2,\mathrm{\dots },x_n$.  The polynomial $Q$ is unique, its coefficients are elements of the ring determined by the coefficients of $P$ and its degree with respect to $p_1,p_2,\mathrm{\dots },p_n$ is same as the degree of $P$ with respect to $x_1$.