A Filtration of the Relative Brauer Group for a Tower of two Quadratic Extensions

Let Formula: see text be a field of characteristic different from Formula: see text, and let Formula: see text be a tower of two quadratic field extensions. We construct a filtration on the relative Brauer group Formula: see text and compute its factors in terms of certain norm groups. The answer depends on whether the extension Formula: see text is biquadratic or not. As a consequence we obtain a description of the group Formula: see text for a Galois field extension Formula: see text with dihedral Galois group. Also, we prove that for any Formula: see text-torsion abelian group Formula: see text and any biquadratic extension Formula: see text there exists a field extension Formula: see text linearly disjoint with Formula: see text such that Formula: see text.