Vector invariant fields of finite classical groups

Let W be an n -dimensional vector space over a finite field F q of any characteristic and mW denote the direct sum of m copies of W . Let F q [ m W ] GL ( W ) and F q ( m W ) GL ( W ) denote the vector invariant ring and vector invariant field respectively where GL ( W ) acts on W in the standard way and acts on mW diagonally. We prove that there exists a set of homogeneous invariant polynomials { f 1 , f 2 , … , f m n } ⊆ F q [ m W ] GL ( W ) such that F q ( m W ) GL ( W ) = F q ( f 1 , f 2 , … , f m n ) . We also prove the same assertions for the special linear groups in any characteristic, and the orthogonal, unitary and symplectic groups in odd characteristic.