Ring of invariants of general linear group over local ring {\mathbb{Z}}_{p^m}

Abstract

Let {\mathbb{Z}}_{p^m} be the ring of integers modulo p^m, where p is a prime and m≥1. The general linear group GL_n({\mathbb{Z}}_{p^m}) acts naturally on the polynomial algebra A_n:={\mathbb{Z}}_{p^m}[x_{\mathrm{1}},\cdots ,x_n]. Denote by A_n^{\mathrm{G}{\mathrm{L}}_n({\mathbb{Z}}_{p^m})} the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.

Key words： Dickson’s theorem; invariant; finite local ring

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