ON THE PROPERTY D(2) AND A COMMON SPLITTING FIELD OF TWO BIQUATERNION ALGEBRAS

摘要

Let F be a field of characteristic ≠ 2. We say that F possesses the property D(2) if for any quadratic extension L/F and any two binary quadratic forms over F having a common nonzero value over L, this value can be chosen in F. There exist examples of fields of characteristic 0 that do not satisfy the property D(2). However, as far as we know, it is still unknown whether there are such examples of positive characteristic and what is the minimal 2-cohomological dimension of fields for which the property D{2) does not hold. In this note it is shown that if k is a field of characteristic ≠ 2 such that ∣k~*/k~(*2)∣ ≥ 4, then for the field k(x) the property D(2) does not hold. Using this fact, we construct two biquaternion algebras over a field K = k(x)((t))((u)) such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e., a field of the kind K(a~(1/2),b~(1/2)), where a, b ∈ K~*) splitting field.