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

Abstract:

Any finite degree field extension $K/F$ determines an ideal ${\mathcal {T}_{K/F}}$ of the Witt ring $WF$ of $F$, called the transfer ideal, which is the image of any nonzero transfer map $WK \to WF$. The ideal ${\mathcal {T}_{K/F}}$ is computed for certain field extensions, concentrating on the case where $K$ has the form $F\left ({\sqrt {{a_1}} , \ldots ,\sqrt {{a_n}} } \right )$, ${a_i} \in F$. When $F$ and $K$ are global fields, we investigate whether there is a local global principle for membership in ${\mathcal {T}_{K/F}}$. 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 ${N_{K/F}}: {K^\ast }/{K^{\ast 2}} \to {F^\ast }/{F^{\ast 2}}$. It is shown that such a Hasse norm theorem holds whenever $K = F(\sqrt {a_1},\ldots ,\sqrt {a_n})$, although it does not always hold for more general extensions of global fields, even some Galois extensions with group $\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z}$.