On the Brauer group complex for a multiquadratic field extension

Let F be a field of characteristic not 2, a1,…,an∈F∗, M=F(a1,…,an). Let further G be the subgroup of F∗/F∗2 generated by a1,…,an. For any I⊂{1,…,n} put aI=∏i∈Iai (a∅=1). Consider the complex∏I≠∅KI∗/KI∗2→φG⊗F∗/F∗2→ψBr2F→resBr2M→∏NM/E∏F⊂E⊂M,[M:E]=2Br2E, where KI=F(aI) are all the quadratic extensions of F, containing in M, φ({wI})=∑IaI⊗NKI/F(wI), ψ(aI⊗z)=(aI,z). The homology group of this complex at i+1-th term from the left is denoted by hi(M/F). Independently of charF we give examples of these complexes with nontrivial h3(M/F) and prescribed n⩾3 and a1,…,an−1. Also similar examples concerning the group h1(M/F) are constructed.