On the product of vector spaces in a commutative field extension

Abstract

Let $K\subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $<AB>$ spanned by the product set $AB=\{ab\mid a\in A,b\in B\}$. If ${\dim }_{K}A=r$ and ${\dim }_{K}B=s$, how small can the dimension of $<AB>$ 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 ${\dim }_K<AB>$ turns out, in this case, to be provided by the numerical function $${\kappa }_{K,L}(r,s)=\underset{h}{\min }(\lceil r/h\rceil +\lceil s/h\rceil -1)h,$$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K\subset H\subset L$. This bound is closely related to one appearing in additive number theory.

arXiv commentSubmitted in November 2007

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