Hurwitz problem
In mathematics, the Hurwitz problem 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.
Description
There are well-known multiplicative relationships between sums of squares in two variables, 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, quaternions, and octonions, respectively.
The Hurwitz problem for the field is to find general relations of the form
with the being bilinear forms in the and : that is, each is a -linear combination of terms of the form.
We call a triple admissible for if such an identity exists. Trivial cases of admissible triples include The problem is uninteresting for 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.
The Hurwitz–Radon theorem
Hurwitz posed the problem in 1898 in the special case and showed that, when coefficients are taken in, the only admissible values were His proof extends to a field of any characteristic except 2.The "Hurwitz–Radon" problem is that of finding admissible triples of the form Obviously is admissible. The Hurwitz–Radon theorem states that is admissible over any field where is the function defined for odd, with and
Other admissible triples include and