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.

Let p be a prime and K a p-adic field (a finite extension of the field of p-adic numbers 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.

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Revised:
Published online:
DOI: 10.5802/jtnb.765
Classification: 17A35,  94B27,  11E76,  11G07
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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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