Power sums over finite subspaces of a field

Abstract

Let V be a finite additive subgroup of a field K of characteristic p > 0. We consider sums of the form Sh(V : α) = Συ∈V (υ + α)h for h ≥ 0 and α ∈ K. In particular, we give necessary and sufficient conditions for the vanishing of Sh(V ; α), in terms of the digit sum in the base-p expansion of h, in the case that V has index p in K. The proof involves the polynomial fV(x) = Πυ∈V(x - υ). We define V† = fV(K). In the case that K is finite, the polynomial fV† can be viewed as a generalised trace TrV : K → V which coincides with the usual trace if V is a subfield of K. © 1999 Academic Press.