# On the product of vector spaces in a commutative field extension

Shalom Eliahou (LMPA), Michel Kervaire, C\'edric Lecouvey (LMPA)

Arxiv ID: 0802.4186•Last updated: 8/19/2021

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 $$ \kappa_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil +
\lceil s/h\rceil -1)h, $$ 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.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.