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

Hurwitz posed the problem in 1898 in the special case ${\displaystyle \;r=s=n\;}$  and showed that, when coefficients are taken in ${\displaystyle \mathbb {C} }$ , the only admissible values ${\displaystyle \,(n,n,n)\,}$  were ${\displaystyle \;n\in \{1,2,4,8\}\;.}$ ^[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 ${\displaystyle \,(r,n,n)\;.}$  Obviously ${\displaystyle \;(1,n,n)\;}$  is admissible. The Hurwitz–Radon theorem states that ${\displaystyle \;\left(\rho (n),n,n\right)\;}$  is admissible over any field where ${\displaystyle \,\rho (n)\,}$  is the function defined for ${\displaystyle \;n=2^{u}v\;,}$  v odd, ${\displaystyle \;u=4a+b\;,}$  with ${\displaystyle \;0\leq b\leq 3\;,}$  and ${\displaystyle \;\rho (n)=8a+2^{b}\;.}$ ^{[1]: 137 [3]: 130}

Other admissible triples include ${\displaystyle \,(3,5,7)\,}$ ^[1]: 138 and ${\displaystyle \,(10,10,16)\;.}$ ^[1]: 137

• Composition algebra
• Hurwitz's theorem (normed division algebras)
• Radon–Hurwitz number