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 (ζv), 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
such that, for all numbers ξ1,…,ξn, η1,…,ηn ∈ ℝ
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.]
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© 1991 Springer Science+Business Media New York
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Koecher, M., Remmert, R. (1991). Composition Algebras. Hurwitz’s Theorem—Vector-Product Algebras. In: Numbers. Graduate Texts in Mathematics, vol 123. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1005-4_13
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DOI: https://doi.org/10.1007/978-1-4612-1005-4_13
Publisher Name: Springer, New York, NY
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