A filtration of the relative Brauer group for a tower of two quadratic extensions

Abstract

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].