Division algebras, Galois fields, quadratic residues (Q1383857)

The structure of division algebras is highlighted from an unconventional point of view. Complex, quaternionic and octonionic products are derived from Galois sequences which can be easily turned into an abelian (additive) group. These sequences, uniquely numbered, can be mapped onto the hyper-complex units, explaining their extraordinary multiplication tables. It is shown that only for the division algebras, a renumbering of basis elements can be found so that the corresponding Galois sequences are equivalent to quadratic residue class codes. Some further remarks are given in various directions. This article should be compared with chapter II.2.6 of \textit{G. M. Dixon} [Division algebras: octanions, quaternions, complex numbers and the design of physics, Kluwer, Dordrecht (1994; Zbl 0807.15024)].