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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, 23 no. 2 (2011), p. 329-346, doi: 10.5802/jtnb.765
Article PDF | Reviews MR 2817933 | Zbl 1242.17006
Class. Math.: 17A35, 94B27, 11E76, 11G07

Résumé - Abstract

Let $p$ be a prime and $K$ a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb{Q}}_p$). 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.

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