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