On the product of vector spaces in a commutative field extension

Let K⊂LK⊂L be a commutative field extension. Given K  -subspaces A,BA,B of L  , we consider the subspace 〈AB〉〈AB〉 spanned by the product set AB={ab|a∈A,b∈B}. If dimKA=rdimKA=r and dimKB=sdimKB=s, how small can the dimension of 〈AB〉〈AB〉 be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on dimK〈AB〉dimK〈AB〉 turns out, in this case, to be provided by the numerical functionκK,L(r,s)=minh(⌈r/h⌉+⌈s/h⌉−1)h, where h runs over the set of K  -dimensions of all finite-dimensional intermediate fields K⊂H⊂LK⊂H⊂L. This bound is closely related to one appearing in additive number theory.