A note about algebras obtained by the Cayley-Dickson process

Abstract

In this paper, we generalize the concepts of level and sublevels of a composition algebra to algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every $t\in \mathbb{N},$ a division algebra $A_t$ of dimension $2^t$ over the power-series field $K\{X_1,X_2,...,X_t\}.$ This gives us the possibility to construct a division algebra of dimension 2${}^t$ and prescribed level 2${}^k$ $k,t\in {\mathbb{N}}^{\ast }.$

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