On the classification of 3-dimensional non-associative division algebras over $p$-adic fields

Abdulaziz Deajim; David Grant

doi:10.5802/jtnb.765

On the classification of 3-dimensional non-associative division algebras over

p

-adic fields

Abdulaziz Deajim ¹; David Grant ²

¹ Department of Mathematics King Khalid University P.O. Box 9004 Abha, Saudi Arabia
² Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395, USA

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 329-346.

Abstract
Résumé

Let $p$ be a prime and $K$ a $p$ -adic field (a finite extension of the field of $p$ -adic numbers $ℚ_{p}$ ). We employ the main results in [12] and the arithmetic of elliptic curves over $K$ to reduce the problem of classifying 3-dimensional non-associative division algebras (up to isotopy) over $K$ to the classification of ternary cubic forms $H$ over $K$ (up to equivalence) with no non-trivial zeros over $K$ . We give an explicit solution to the latter problem, which we then relate to the reduction type of the jacobian of $H$ .

This result completes the classification of 3-dimensional non-associative division algebras over number fields done in [12]. These algebras are useful for the construction of space-time codes, which are used to make communications over multiple-transmit antenna systems more reliable.

Soient p un nombre premier et $K$ un corps $p$ -adique. On emploie les résultats de [12] et l’arithmétique des courbes elliptiques sur $K$ pour réduire le problème de classification des algèbres à division non associatives de dimension 3 sur $K$ à celui de la classification des formes cubiques ternaires $H$ sur $K$ sans zéros non-triviaux. On donne une solution explicite du dernier problème qu’on relie ensuite à la réduction de la jacobienne de $H$ .

Ce résultat complète la classification des algèbres à division non associatives de dimension 3 sur les corps de nombres faite dans [12]. Ces algèbres sont utiles pour la construction des codes espace-temps utilisés pour une meilleure fiabilité des communications à travers les systèmes multi-antennes.

MR Zbl

DOI: 10.5802/jtnb.765

Classification: 17A35, 94B27, 11E76, 11G07

Author's affiliations:

Abdulaziz Deajim ¹; David Grant ²

¹ Department of Mathematics King Khalid University P.O. Box 9004 Abha, Saudi Arabia
² Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395, USA

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     title = {On the classification of 3-dimensional non-associative division algebras over $p$-adic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {329--346},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
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Abdulaziz Deajim; David Grant. On the classification of 3-dimensional non-associative division algebras over $p$-adic fields. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 329-346. doi : 10.5802/jtnb.765. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.765/

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