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

Abdulaziz Deajim; David Grant — 2011

Journal de Théorie des Nombres de Bordeaux

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...

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

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