Levels and sublevels of division algebras obtained by the Cayley-Dickson process

TL;DR

This paper extends the concepts of level and sublevel to division algebras created via the Cayley-Dickson process, demonstrating the construction of division algebras with specific dimensions, levels, and sublevels over certain fields.

Contribution

It generalizes the notions of level and sublevel to a broader class of division algebras obtained through the Cayley-Dickson process, providing explicit constructions with prescribed properties.

Findings

Constructs division algebras with prescribed levels and sublevels

Shows existence of division algebras of dimension 2^t with specific properties

Extends Brown's construction to more general settings

Abstract

{\small \ We generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to division algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in \mathbb{N},${\small \ a nonassociative division algebra}$A_{t}${\small \ of dimension}$2^{t}${\small \ over the power-series field}$K\{X_{1},X_{2},...,X_{t}\}.\,${\small \ This gives us the possibility to construct a division algebra of \thinspace dimension}$% {\small \,2}^{t}\,${\small and prescribed \thinspace level and sublevel}$% {\small \,\,2}^{k}${\small ,\thinspace \thinspace}$k,\,t\in \mathbb{N}-\{0\} $ and dimension ${\small \,2}^{t}\,${\small and prescribed \thinspace level}${\small \,\,2}^{k}+1${\small ,\thinspace \thinspace}$k\in \mathbb{N}% -\{0\},t\in \mathbb{N},t\geq 2.$