  Let k be an algebraically closed field of characteristic p > 0. Let H be a
subgroup of GL(n,k). We are interested in the determination of the vector
invariants of H. When the characteristic of k is 0, it is known that the
invariants of d vectors, d > n, are obtained from those of n vectors by
polarization. This result is not true when char k = p > 0 even in the case
where H is a torus. However, we show that the algebra of invariants is always
integral over the algebra of polarized invariants and when H is reductive is
actually the p - root closure of that algebra. We also give conditions for the
algebras to be equal, relating equality to good filtrations and saturated
subgroups. We conclude with examples where H is finite or a classical group or
is a certain kind of unipotent subgroup of GL(n,k).
