On the product of vector spaces in a commutative field extension - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Journal of Number Theory Année : 2009

On the product of vector spaces in a commutative field extension

Résumé

Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace $\langle AB \rangle$ 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 $\langle AB \rangle$ 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 $\dim_K \langle AB \rangle$ 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.
Fichier principal
Vignette du fichier
extensions.pdf (151.79 Ko) Télécharger le fichier
Origine Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-00259373 , version 1 (27-02-2008)

Identifiants

Citer

Shalom Eliahou, Michel Kervaire, Cédric Lecouvey. On the product of vector spaces in a commutative field extension. Journal of Number Theory, 2009, 129 (2), pp.339-348. ⟨10.1016/j.jnt.2008.06.004⟩. ⟨hal-00259373⟩
86 Consultations
74 Téléchargements

Altmetric

Partager

Gmail Mastodon Facebook X LinkedIn More