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Gro-Tsen
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An isomorphic classification of non-associative division octonion algebras

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and $N(x)\ne 0$ for every non-zero elements $x,y\in A$.

Wikipedia writes that two division octonion algebras $(A,N)$ and $(A',N')$ over a field $F$ are $F$-isomorphic if and only if their norms $N$, $N'$ are $F$-isomorphic. This reduces the problem of isomorphic classification of division octonion algebras to the problem of classification of their norms. Is there any additional information on such a classification?

Question. What is the number of non-isomorphic division octonion algebras over a given field? Can it be greater than one? Can it be infinite?

Because finite or algebraically closed fields admit no division octonion algebras, see the lecture notes of Konrad Voelkel.

Taras Banakh
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