Polynomial invariants of finite groups over fields of prime characteristics

Abstract

Let R be a commutative ring with the unit element 1, and let G = Sn be the symmetric \ngroup of degree n 2'. 1. Let A~n denote the subalgebra of invariants of the polynomial algebra Arnn = \nR[x11 , ... ,x1n; ... ;xm1, ... ,Xmn] with respect to G. A classical result of Noether [6] implies that if \nevery non-zero integer is invertible in R, then A~n is generated by polarized elementary symmetric \npolynomials. As was recently shown by D. Richman, this result remains true under the condition that \nn! is invertible in R. The purpose of this paper is to give a short proof of Richman's result based on \nthe use of Waring's formula and closely related t0 Noether's original proof.