In mathematics, theHurwitz problem (named afterAdolf Hurwitz) is the problem of finding multiplicative relations betweenquadratic 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

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

(known as theBrahmagupta–Fibonacci identity), and alsoEuler's four-square identity andDegen's eight-square identity. These may be interpreted as multiplicativity for the norms on thecomplex numbers (${\displaystyle \mathbb{C}}$),quaternions (${\displaystyle \mathbb{H}}$), andoctonions (${\displaystyle \mathbb{O}}$), respectively.^{[1]: 1–3 [2]}

The Hurwitz problem for the fieldK 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 thez being bilinear forms in thex andy: that is, eachz is aK-linear combination of terms of the formx_i y_j.^[3]: 127

We call a triple${\displaystyle (r,s,n)}$admissible forK if such an identity exists.^[1]: 125 Trivial cases of admissible triples include${\displaystyle (r,s,rs).}$ The problem is uninteresting forK ofcharacteristic 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 anycharacteristicexcept 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. TheHurwitz–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${\displaystyle 0\le b\le 3,}$ 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

• Field (mathematics)
• Quadratic forms
• Mathematical problems