Mélanie Raczek: On ternary cubic forms that determine central simple algebras of degree 3


Submission: 2008, Nov 24

Fixing a field F of characteristic different from 2 and 3, we consider pairs (A,V) consisting of a degree 3 central simple F-algebra A and a 3-dimensional vector subspace V of the reduced trace zero elements of A which is totally isotropic for the trace quadratic form. Each such pair gives rise to a cubic form mapping an element of V to its cube; therefore we call it a cubic pair over F. Using the Okubo product in the case where F contains a primitive cube root of unity, and extending scalars otherwise, we give an explicit description of all isomorphism classes of such pairs over F. We deduce that a cubic form associated with an algebra in this manner determines the algebra up to (anti-)isomorphism.

2000 Mathematics Subject Classification: 11E76, 12E15, 12E15, 14H45

Keywords and Phrases: Forms of degree higher than two, Skew fields, division rings, Special curves and curves of low genus

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