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An analogue of Vosper's theorem for extension fields

Published online by Cambridge University Press:  31 January 2017

CHRISTINE BACHOC ,
ORIOL SERRA  and
GILLES ZÉMOR
CHRISTINE BACHOC
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: bachoc@math.u-bordeaux.fr
ORIOL SERRA
Affiliation:
Departament de Matemàtiques, Universitat Politècnica de Catalunya and Barcelona Graduate School of Mathematics, Edifici C3, Despatx: 112, C. Jordi Girona, 1–3, 08034 Barcelona, Spain. e-mail: oriol.serra@upc.edu
GILLES ZÉMOR
Affiliation:
Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux, 351 cours de la Libération, 33400 Talence, France. e-mail: zemor@math.u-bordeaux.fr
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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

\begin{linenomath}$$ \dim_FST =\dim_F S+\dim_F T-1, $$\end{linenomath}
when dimFS, dimFT ⩾ 2 and dimFST ⩽ [L : F] − 2.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 423 - 452
Copyright
Copyright © Cambridge Philosophical Society 2017 

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