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

Abstract

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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.

How to cite

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Deajim, 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 -

References

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