An analogue of Vosper's Theorem for Extension Fields

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 ${\dim }_{F}ST={\dim }_{F}S+{\dim }_{F}T-1,$ when ${\dim }_{F}S,{\dim }_{F}T\ge 2$ and ${\dim }_{F}ST\le [L:F]-2$.

33 pages

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