A COUNTEREXAMPLE TO THE HASSE PRINCIPLE FOR SIMILARITIES OF BILINEAR-FORMS

Let k be a number field and V a vector space of finite dimension over k. We first translate the Hasse principle for similarities of non symmetric bilinear forms to a Hasse principle relative to norms for some special extensions of k. Then we give a counterexample to this new principle coming from a Galois extension of Q with Galois group the quaternion group of order 8.