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Hurwitz’s theorem

Theorem: Except for typeset structure, there exists no identity

(x _ 1^2 + x _ 2^2 + ... + x _ n^2) (y _ 1^2 + y _ 2^2 + ... + y _ n^2) = z _ 1^2 + z _ 2^2 + ... + z _ n^2(1)

expressing the product typeset structure as a sum of the squares on typeset structure bilinear functions of  typeset structure  and  typeset structure.

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  typeset structure is a field with characteristic not 2, the sum of squares identity of the form (1)
where each typeset structure is a rational function of  typeset structurei and typeset structure is possible if and only if typeset structure is a power of  2 .  

References

[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.

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