Nonassociative cyclic algebras (Q2017167)

Starting from previous results, in this paper, the author generalises nonassociative quaternion algebras and introduces nonassociative cyclic algebras of degree \(n\), taking the element \(a\) to be outside of the base field \(F\). As a particular case, he considers nonassociative cyclic algebras whith \(L/F\) a cyclic field extension of degree \(3\). He proves that all of these algebras are division algebras for all \(a\in L\backslash F\). This result is generalised for nonassociative cyclic algebras of degree \(n\). He proves that if \(n\) is prime, then a nonassociative cyclic algebra is a division algebra for all \(a\in L\backslash F\). If \(n\) is not prime, if the elements \( \{1,a,a^2,\ldots,a^{n-1}\}\) are linearly independent over \(F\), then the nonassociative cyclic algebras of degree \(n\) is with division.