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Let F be a field of characteristic different from 2, and let L = F(b + 2ca) be a tower of two quadratic field extensions. We construct a filtration on the relative Brauer group Br(L/F) and compute its factors in terms of certain norm groups. The answer depends on whether the extension L/F is biquadratic or not. As a consequence we obtain a description of the group F∗2N E/FE∗ for a Galois field extension E/F with dihedral Galois group. Also, we prove that for any 2-torsion abelian group G and any biquadratic extension L/F there exists a field extension K/F linearly disjoint with L/F such that Br(LK/K)/2Br(LK/K) ≃ G. [ABSTRACT FROM AUTHOR]

Abstract: The homology groups h 1 ( l / k ), h 2 ( l / k ), and h 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 in h 2 ( l / k ) to be zero. We consider as well the second divided power operation γ 2 : 2 Br ( l / k ) → H 4 ( k , Z / 2 Z ), and show that there are nonstandard elements with respect to γ2. Further, a natural transformation h 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 group h 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 nontrivial h 3 ( L / F ) are given. As a consequence we show that the homology group h 3 ( L / F ) can be arbitrarily big. [ABSTRACT FROM AUTHOR]