# 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 (2011)

- Volume: 23, Issue: 2, page 329-346
- ISSN: 1246-7405

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topDeajim, Abdulaziz, and Grant, David. "On the classification of 3-dimensional non-associative division algebras over $p$-adic fields." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 329-346. <http://eudml.org/doc/219755>.

@article{Deajim2011,

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.},

affiliation = {Department of Mathematics King Khalid University P.O. Box 9004 Abha, Saudi Arabia; Department of Mathematics University of Colorado at Boulder Boulder, Colorado 80309-0395, USA},

author = {Deajim, Abdulaziz, Grant, David},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {division algebra; p-adic field},

language = {eng},

month = {6},

number = {2},

pages = {329-346},

publisher = {Société Arithmétique de Bordeaux},

title = {On the classification of 3-dimensional non-associative division algebras over $p$-adic fields},

url = {http://eudml.org/doc/219755},

volume = {23},

year = {2011},

}

TY - JOUR

AU - Deajim, Abdulaziz

AU - Grant, David

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

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2011/6//

PB - Société Arithmétique de Bordeaux

VL - 23

IS - 2

SP - 329

EP - 346

AB - 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.

LA - eng

KW - division algebra; p-adic field

UR - http://eudml.org/doc/219755

ER -

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