| Abstract: | The homology groups  ,  , and  of the Brauer complex for a triquadratic field extension  are studied. In particular, given  , we find equivalent conditions for the image of D in  to be zero. We consider as well the second divided power operation  , and show that there are nonstandard elements with respect to γ2. Further, a natural transformation  , which turns out to be nondegenerate on the left, is defined. As an application we construct a field extension  such that the cohomology group  of the Brauer complex contains the images of prescribed elements of  , provided these elements satisfy a certain cohomological condition. At the final part of the paper examples of triquadratic extensions  with nontrivial  are given. As a consequence we show that the homology group  can be arbitrarily big. |
