algebraic independence of elementary symmetric polynomials

Theorem.

Let $s_{\mathrm{1}}\mathrm{,}{s}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{s}_n$ be the elementary symmetric polynomials in $n$ variables $t_{\mathrm{1}}\mathrm{,}{t}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{t}_n$ over a commutative ring $R$.Then $s_{\mathrm{1}}\mathrm{,}{s}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{s}_n$ are algebraically independent elements of$R\mathit{}\mathrm{[}{t}_{\mathrm{1}}\mathrm{,}{t}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{t}_n\mathrm{]}$ over $R$.