Hurwitz problem

There are well-known multiplicative relationships between sums of squares in two variables

${\displaystyle (x^{2}+y^{2})(u^{2}+v^{2})=(xu-yv)^{2}+(xv+yu)^{2}\ ,}$

(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 (${\displaystyle \mathbb {C} }$), quaternions (${\displaystyle \mathbb {H} }$), and octonions (${\displaystyle \mathbb {O} }$), respectively.^{[1]: 1–3 [2]}

The Hurwitz problem for the field K is to find general relations of the form

${\displaystyle (x_{1}^{2}+\cdots +x_{r}^{2})\cdot (y_{1}^{2}+\cdots +y_{s}^{2})=(z_{1}^{2}+\cdots +z_{n}^{2})\ ,}$

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 x_i y_j.^[3]: 127

We call a triple ${\displaystyle \;(r,s,n)\;}$ admissible for K if such an identity exists.^[1]: 125 Trivial cases of admissible triples include ${\displaystyle \;(r,s,rs)\;.}$ 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