On the Brauer group complex for a multiquadratic field extension

Let F be a field of characteristic not 2, a 1 , … , a n ∈ F ∗ , M = F ( a 1 , … , a n ) . Let further G be the subgroup of F ∗ / F ∗ 2 generated by a 1 , … , a n . For any I ⊂ { 1 , … , n } put a I = ∏ i ∈ I a i ( a ∅ = 1 ) . Consider the complex ∏ I ≠ ∅ K I ∗ / K I ∗ 2 → φ G ⊗ F ∗ / F ∗ 2 → ψ Br 2 F → res Br 2 M → ∏ N M / E ∏ F ⊂ E ⊂ M , [ M : E ] = 2 Br 2 E , where K I = F ( a I ) are all the quadratic extensions of F , containing in M , φ ( { w I } ) = ∑ I a I ⊗ N K I / F ( w I ) , ψ ( a I ⊗ z ) = ( a I , z ) . The homology group of this complex at i + 1 -th term from the left is denoted by h i ( M / F ) . Independently of char F we give examples of these complexes with nontrivial h 3 ( M / F ) and prescribed n ⩾ 3 and a 1 , … , a n − 1 . Also similar examples concerning the group h 1 ( M / F ) are constructed.