On products of additive functions (a third approach)

Summary

Leta ₁, ⋯, a_s : G → K be additive functions from an abelian groupG into a fieldK such thata ₁(g)·⋯·a_s(g) = 0 for allg ∈ G. If char(K) =0, then it is well known that one of the functions a₁ has to vanish. We give a new proof of this result and show that, if char(K) > 0, it is only valid under additional assumptions.