The Hopf condition for bilinear forms over arbitrary fields

Abstract

We settle an old question about the existence of certain ‘sums-of-squares’ formulas over a field $F$, related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of characteristic $0$, then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic $p>2$.