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Theorem: Except for , there exists no identity
(1) |
expressing the product as a sum of the squares on bilinear functions of and .
Hurwitz [1] proved this result with the help of matrix multiplication. A more readable form of Hurwitz proof can be found in [2] .
Note that in 1967 A.Pfister [3] proved that if is a field with characteristic not 2, the sum of squares identity of the form (1)
where each is a rational function of i and is possible if and only if is a power of 2 .
[1] | Hurwitz, A. (1898). Ueber die Composition der quadratischen Formen von beliebig vielen Variabeln. Gött. Nachr., 309-316. |
[2] | Dickson, L. E. (1919). On quaternions and their generalization and the history of the eight square theorem. Annals of Math., (2) 20, 155-171, 297. |
[3] | Pfister, A. (1967). Zur Darstellung definiter Funktionen als Summe von Quadraten. (To the representation of definite functions as the sum of squares). (German). Invent. Math., 4, 229-237. |
Cite this web-page as:
Štefan Porubský: Hurwitz theorem.