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

The homology groups h1(l/k), h2(l/k), and h3(l/k) of the Brauer complex for a triquadratic field extension l=k(a,b,c) are studied. In particular, given D∈APTARAMOPREFIX2 Br (k(a,b,c)/k), we find equivalent conditions for the image of D in h2(l/k) to be zero. We consider as well the second divided power operation γ2:APTARAMOPREFIX2 Br (l/k)→H4(k,Z/2Z), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h2⊗h1→H3, 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 group h1(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 nontrivial h3(L/F) are given. As a consequence we show that the homology group h3(L/F) can be arbitrarily big.