A Note on the Classical Dickson Invariants.

Let 픽q 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 $[e_1,e_2,\dots,e_n]:=\det (x_j^{q^{e_i}})_{n\times n} \in {\Bbb F}_q[x_1,x_2,\dots,x_n]$, where e1,..., en are non-negative integers. He observed that any [e1,..., en] is divisible by [0,1,...,n-1], and the quotient is GLn(픽q)-invariant. He then went on to show that one can pick n of these invariants to generate the entire invariant subring of GLn(픽q). In this paper, we answer the following two natural questions: How to express any given [e1,..., en]/[0,1,...,n-1] in terms of the fundamental generators? In general, when does [e1,..., en] divide [f1,..., fn]? The answers are Theorems 1.2 and 1.3, respectively. [ABSTRACT FROM AUTHOR]