On the homology groups of the Brauer complex for a triquadratic field extension

The homology groupsh 1 ( l / k ) ,h 2 ( l / k ) , andh 3 ( l / k )of the Brauer complex for a triquadratic field extension l = k ( a , b , c )are studied. In particular, given D ∈2 Br( k ( a , b , c ) / k ) , we find equivalent conditions for the image of D inh 2 ( l / k )to be zero. We consider as well the second divided power operationγ 2 :2 Br( l / k ) → H 4 ( k , Z / 2 Z ) , and show that there are nonstandard elements with respect to γ 2 . Further, a natural transformationh 2 ⊗ h 1 → H 3 , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension F / k such that the cohomology grouph 1( F ( a , b , c ) / F )of the Brauer complex contains the images of prescribed elements of k ∗ , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions L / F with nontrivialh 3 ( L / F )are given. As a consequence we show that the homology grouph 3 ( L / F )can be arbitrarily big.