An analogue of Vosper's Theorem for Extension Fields

TL;DR

This paper establishes a linear analogue of Vosper's Theorem for field extensions, characterizing pairs of subspaces with small product span dimension in prime extensions of finite fields.

Contribution

It introduces a new theorem that describes the structure of subspaces with minimal product span dimension in prime finite field extensions.

Findings

Characterization of subspace pairs with small product dimension

Extension of Vosper's Theorem to linear algebra context

Conditions for minimal product span in prime extensions

Abstract

We are interested in characterising pairs $S,T$ of $F$-linear subspaces in a field extension $L/F$ such that the linear span $ST$ of the set of products of elements of $S$ and of elements of $T$ has small dimension. Our central result is a linear analogue of Vosper's Theorem, which gives the structure of vector spaces $S, T$ in a prime extension $L$ of a finite field $F$ for which $\dim_FST =\dim_F S+\dim_F T-1,$ when $\dim_F S, \dim_F T\ge 2$ and $\dim_F ST\le [L:F]-2$.