Power Sums over Finite Subspaces of a Field

LetVbe a finite additive subgroup of a fieldKof characteristicp>0. We consider sums of the formSh(V:α)=∑v V(v+α)hforh≥0 and α K. In particular, we give necessary and sufficient conditions for the vanishing ofSh(V; α), in terms of the digit sum in the base-pexpansion ofh, in the case thatVhas indexpinK. The proof involves the polynomialfV(x)=∏v V(x−v). We defineV†=fV(K). In the case thatKis finite, the polynomialfV†can be viewed as a generalised trace TrV:K→Vwhich coincides with the usual trace ifVis a subfield ofK.