Symmetric polynomials over finite fields

It is shown that two vectors with coordinates in the finite q-element field of characteristic p belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree pk,2pk,…,(q−1)pk, k=0,1,2,… has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when q=p and the dimension is large compared to p. A relatively small separating set of multisymmetric polynomials over the field of q elements is derived.