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

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. Introduction Consider a field F of characteristic different from 2. Let A be a quaternion algebra over F and let A denote the subspace of reduced trace zero elements of A. Then for all x ∈ A we have x ∈ F . We thus obtain a quadratic form on A mapping x to x. Up to the sign, this quadratic form is the norm form of the quaternion algebra restricted to A. By Theorem 2.5, p. 57, in [Lam, 2005], this quadratic form determines the quaternion algebra up to isomorphism. In this paper we shall generalize this construction for algebras of degree 3. Consider a field F of characteristic different from 2 and 3 and let A be a degree 3 central simple algebra over F . Again let A denote the subspace of reduced trace zero elements of A. Then the cube of an arbitrary element x ∈ A need not be in F in general. In fact, it is in F if and only if the reduced trace of x is equal to zero. Let q : A → F be the trace quadratic form on A (mapping x to the reduced trace of x). Then the Witt index of q is equal to 4 if F contains a primitive cube root of unity and is equal to 3 otherwise (see Lemma 0.1). In both cases there exist 3-dimensional subspaces of A which are totally isotropic for the trace quadratic form. Each such vector subspace V ⊂ A gives rise to