On the product of vector spaces in a commutative field extension

TL;DR

This paper determines the minimal possible dimension of the span of the product of two subspaces in a field extension, providing a complete characterization in characteristic zero and for separable extensions.

Contribution

It introduces a precise lower bound for the dimension of the product span, linking it to intermediate field dimensions and extending additive number theory concepts.

Findings

Established the exact lower bound for characteristic zero and separable extensions.

Connected the bound to a numerical function involving intermediate fields.

Provided a complete answer to the minimal dimension problem in the specified setting.

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