The transfer ideal of quadratic forms and a hasse norm theorem mod squares

Any finite degree field extension K/F determines an ideal ^Ik/f of the Witt ring WF of F, called the transfer ideal, which is the image of any nonzero transfer map WK → WF. The ideal ^Ik/f is computed for certain field extensions, concentrating on the case where K has the form F(√ a₁;,&, √ a_n), a_iε F. When F and K are global fields, we investigate whether there is a local global principle for membership in ^Ik/f. This is shown to be equivalent to the existence of a "Hasse norm theorem mod squares, " i.e., a local global principle for the image of the norm map N_K/F: KK²→ F/F2. It is shown that such a Hasse norm theorem holds whenever K = F(√ a₁;,&, √ a_n), although it does not always hold for more general extensions of global fields, even some Galois extensions with group Z/2Z × Z/4Z.