Composition Algebras. Hurwitz’s Theorem—Vector-Product Algebras

Abstract

1. For multiplication in the algebras ℝ, ℂ, ℍ and O, the formula |xy|² = |x|²|y|², holds, where | | denotes the Euclidean length. If one expresses the vectors x, y and z := xy in terms of their coordinates with respect to an orthonormal basis, as (ξ _v ), (η _v ), and (ζ_v), respectively, then we obtain, in view of the bilinearity of the product xy the Squares Theorem. In the four cases n = 1,2,4,8 there are n real (in fact rational integral) bilinear forms

$$ \zeta _v = \sum\limits_{\lambda ,\mu = 1}^n {\alpha _{\lambda \mu }^{(v)} \xi _\lambda \eta _\mu ,\,\alpha _{\lambda \mu }^{(v)} \in \mathbb{Z},\,v = 1, \ldots ,n} $$

such that, for all numbers ξ₁,…,ξ_n, η₁,…,η_n ∈ ℝ

$$ \zeta _1^2 + \zeta _2^2 + \cdots + \zeta _n^2 = \left( {\xi _1^2 + \xi _2^2 + \cdots + \xi _n^2 } \right)\left( {\eta _1^2 + \eta _2^2 + \cdots + \eta _n^2 } \right) $$

((*))

Durch diesen Nachweis wird die alte Streitfrage, ob sich die bekannten Produktformeln für Summen von 2, 4 und 8 Quadraten auf Summen von mehr als 8 Quadraten ausdehnen lassen, endgültig, und zwar in verneinendem Sinne entschieden (A. Hurwitz 1898).

[By this proof, the long-debated question of whether the well-known product formulae for sums of 2, 4 and 8 squares can be extended to more than 8 squares, has finally been answered in the negative.]