Cristina Flaut and Mirela Stefanescu: Some equations over generalized quaternion and octonion division algebras, p.427-439

Abstract:

It is known that any polynomial of degree $n$ with coefficients in a field $K$ has at most $n$ roots in $K.$ If the coefficients are in $\Bbb{H}$ (the quaternion algebra), the situation is different. For $\Bbb{H}$ over the real field, there is a kind of a fundamental theorem of algebra: If a polynomial has only one term of the greatest degree then it has at least one root in $\Bbb{H}.$ A similar theorem is also true for the octonions.

In this paper we try to solve, in general or in particular cases, some quadratic and linear equations with two different terms of greatest degree and the coefficients in the generalized division quaternion and octonion algebras $\Bbb{H}\left( \alpha ,\beta \right)$ and $% \Bbb{O(}\alpha ,\beta ) \,$over an arbitrary field $K,\,\,charK\neq 2.$

Key Words: Quaternion algebra; Division algebra; Octonion algebra.

2000 Mathematics Subject Classification: Primary: 17D05,
Secondary: 17D99.

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