Stronger division criteria in the cubic Menichetti algebra case kk ∈ F×

Derive stronger sufficient conditions ensuring that the Menichetti algebra (K/F,a,b,c), constructed from a cubic Galois field extension K/F with parameters a,b,c in K×, is a division algebra over F in the special case when kk = b^{-2} c a^{-1} belongs to F×, beyond the criteria provided in Theorem 4.5.

Background

The paper studies nonassociative Menichetti algebras (K/F,k0,...,km−1), focusing on division criteria when K/F is an abelian Galois extension. In Section 4, the authors analyze the cubic case (K/F,a,b,c), deriving determinant formulas and sufficient conditions for the algebra to be a division algebra.

Theorem 4.5 provides conditions based on norms and linear independence of ca−1 and b−1c. However, in a specific subcase—when kk equals b^{-2} c a^{-1} and lies in F×—the authors note they cannot obtain stronger conditions. Establishing improved criteria in this scenario would sharpen the understanding of when these cubic Menichetti algebras are division algebras.