In mathematics, the Hurwitz problem (named after Adolf Hurwitz) is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables.
There are well-known multiplicative relationships between sums of squares in two variables (known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity.
These may be interpreted as multiplicativity for the norms on the complex numbers (
[1]: 1–3 [2] The Hurwitz problem for the field K is to find general relations of the form with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xi yj.
admissible for K if such an identity exists.
[1]: 125 Trivial cases of admissible triples include
The problem is uninteresting for K of characteristic 2, since over such fields every sum of squares is a square, and we exclude this case.
It is believed that otherwise admissibility is independent of the field of definition.
[1]: 137 Hurwitz posed the problem in 1898 in the special case
and showed that, when coefficients are taken in
[3]: 130 His proof extends to a field of any characteristic except 2.
[1]: 3 The "Hurwitz–Radon" problem is that of finding admissible triples of the form
The Hurwitz–Radon theorem states that
is the function defined for
[1]: 137 [3]: 130 Other admissible triples include