THE LENGTHS OF THE QUATERNION AND OCTONION ALGEBRAS

摘要

The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (R), complex numbers (C), quaternions (H), and octonions (O). The length of R as an algebra over itself is zero; the length of C as an R-algebra equals one. The purpose of the present paper is to prove that the lengths of the R-algebras of quaternions and octonions equal two and three, respectively. Bibliography: 8 titles.