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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The transfer ideal of quadratic forms and a Hasse norm theorem mod squares
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by David B. Leep and Adrian R. Wadsworth PDF
Trans. Amer. Math. Soc. 315 (1989), 415-432 Request permission

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 $\mathbb {Z}/2\mathbb {Z} \times \mathbb {Z}/4\mathbb {Z}$.
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • 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