Levels and sublevels of algebras obtained by the Cayley–Dickson process

Abstract

In this paper, we generalize the concepts of level and sublevel of a composition algebra to algebras obtained by the Cayley–Dickson process and we will show that, in the case of level for algebras obtained by the Cayley–Dickson process, the situation is the same as for the integral domains, proving that for any positive integer n, there is an algebra A obtained by the Cayley–Dickson process with the norm form anisotropic over a suitable field, which has the level \({n \in \mathbb{N}-\{0\}}\) .