Abstract
Let F q be the finite field with q elements, where q is a power of some prime p. In the classical paper [2], Dickson defined the polynomial [e 1 ,e 2 ,...,e n ] :=det (x j qe i )n×n ∈ F q [x 1 ,x 2 ,...,x n ], where e 1 ,..., e n are non-negative integers. He observed that any [e 1 ,..., e n ] is divisible by [0,1,...,n-1], and the quotient is GL n (F q )-invariant. He then went on to show that one can pick n of these invariants to generate the entire invariant subring of GL n (F q ). In this paper, we answer the following two natural questions: How to express any given [e 1 ,..., e n ]/[0,1,..., n-1] in terms of the fundamental generators? In general, when does [e 1 ,..., e n ] divide [f 1 ,..., f n ]? The answers are Theorems 1.2 and 1.3, respectively. © 2010 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.