Degree four cohomological invariants for certain central simple algebras

Abstract

Let F be a field, charF≠2. In the first section of the paper we prove that if A=(a,b)+(c,d) is a biquaternion algebra divisible by 2 in the Brauer group Br(F), and 《−1,−1》F=0, then the symbol (a,b,c,d)∈H4(F,Z/2Z) is an invariant, i.e. it does not depend on the decomposition of A into a sum of two quaternions. In the second section we construct an invariant p in H4(F,Z/2Z) for elements C+α∈4Br(F), where C is cyclic of degree at most 4, and α∈2Br(F). In the case −1∈F⁎ we extend the invariant p to elements D+α∈4Br(F), where indD≤4 and α∈2Br(F).