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 $R$.

Title	algebraic independence of elementary symmetric polynomials
Canonical name	AlgebraicIndependenceOfElementarySymmetricPolynomials
Date of creation	2013-03-22 14:49:11
Last modified on	2013-03-22 14:49:11
Owner	mclase (549)
Last modified by	mclase (549)
Numerical id	5
Author	mclase (549)
Entry type	Theorem
Classification	msc 05E05