Quartic polynomials and the Hasse norm theorem modulo squares

Let F   be a field, charF≠2, L/FL/F a quartic field extension. Define by GL/FGL/F the group of elements r∈F⁎r∈F⁎ such that D∪(r)=0D∪(r)=0 for any regular field extension K/FK/F and any D∈Br2(KL/K). We show that GL/F=F⁎2NL/FL⁎GL/F=F⁎2NL/FL⁎. As a consequence we prove that the Hasse norm theorem modulo squares holds for L/FL/F.