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Any finite degree field extension K/F determines an ideal JK/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 59K/F is computed for certain field extensions, concentrating on the case where K has the form F(Va-,... , a), ai E F. When F and K are global fields, we investigate whether there is a local global principle for membership in TKIFThis 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 NK/F: K*/K*2 F*/F*2. It is shown that such a Hasse norm theorem holds whenever K = F(.aj, . . ., Va'), although it does not always hold for more general extensions of global fields, even some Galois extensions with group Z/2Z x Z/4Z.

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