On Modular Vector Invariant Fields

Let F_(q)be a finite field of any characteristic and GL(n,F_(q))be the general linear group over F_(q).Suppose W denotes the standard representation of GL(n,F_(q)),and GL(n,F_(q))acts diagonally on the direct sum of W and its dual space W^(∗).Let G be any subgroup of GL(n,F_(q)).Suppose the invariant field F_(q)(W)G=F_(q)(f1,f2,…,fk),where f1,f2,…,fk in F_(q)[W]G are homogeneous invariant polynomials.We prove that there exist homogeneous polynomialsl1,l2,…,ln in the invariant ring F_(q)[W⊕W^(∗)]G such that the invariant field F_(q)(W⊕W^(∗))G is generated by{f1,f2,…,fk,l1,l2,…,ln}over F_(q).