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.

This paper uses a relative of B P-cohomology to prove a theorem in characteristic p algebra. Specifically, we obtain some new necessary conditions for the existence of sumsof-squares formulas over fields of characteristic p > 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalizedétale cohomology theory calledétale B P2.

We define a new invariant in the homology of a differential graded algebra. This invariant is the obstruction to defining a fourfold Massey product. It can be used to detect differential graded algebras that are not quasiisomorphic. We also make an explicit calculation in the cohomology of the Steenrod algebra.