Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: David B. Leep and Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 315 (1989), 415-432
MSC: Primary 11E81; Secondary 11E12, 11R37
DOI: https://doi.org/10.1090/S0002-9947-1989-0986030-X
MathSciNet review: 986030
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z}$.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11E81, 11E12, 11R37

Retrieve articles in all journals with MSC: 11E81, 11E12, 11R37


Additional Information

Article copyright: © Copyright 1989 American Mathematical Society