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Transactions of the American Mathematical Society
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
MathSciNet review: 986030
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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^{\ast2}} \to {F^\ast}/{F^{\ast2}}$. 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}$.

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PII: S 0002-9947(1989)0986030-X
Article copyright: © Copyright 1989 American Mathematical Society

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