Polynomial invariants of finite groups over fields of prime characteristic

Let $R$ be a commutative ring with the unit element $1$, and let $G=S_n$ be the symmetric group of degree $n \geq 1$. Let $A_{mn}^G$ denote the subalgebra of invariants of the polynomial algebra $A_{mn}=R[x_{11},\ldots,x_{1n};\ldots;x_{m1},\ldots,x_{mn}]$ with respect to $G$. A classical result of Noether [6] implies that if every non-zero integer is invertible in $R$, then $A_{mn}^G$ is generated by polarized elementary symmetric polynomials. As was recently shown by D. Richman, this result remains true under the condition that $n!$ is invertible in $R$. The purpose of this paper is to give a short proof of Richman's result based on the use of Waring's formula and closely related to Noether's original proof.
The research was supported by Bilkent University, 06533 Bilkent, Ankara, Turkey.